The Booms and Busts of Beta Arbitrage*

Shiyang Huang University of Hong Kong

Email: huangsy@hku.hk

Dong Lou London School of Economics and CEPR

Email: d.lou@lse.ac.uk

Christopher Polk London School of Economics and CEPR

Email: c.polk@lse.ac.uk

This Draft: March 2018

* We would like to thank Nicholas Barberis, Sylvain Champonnais, Andrea Frazzini, Pengjie Gao,

Emmanuel Jurczenko, Ralph Koijen, Toby Moskowitz, Lasse Pedersen, Steven Riddiough, Emil

Siriwardane, Dimitri Vayanos, Michela Verardo, Tuomo Vuolteenaho, Yu Yuan, and seminar participants

at Arrowstreet Capital, Citigroup Quant Research Conference, London Quant Group, London Business

School, London School of Economics, Norwegian School of Economics, Renmin University, UBS Quant

Conference, University of Cambridge, University of California in Los Angeles, University of Rotterdam,

University of Warwick, 2014 China International Conference in Finance, 2014 Imperial College Hedge Fund

Conference, 2014 Quantitative Management Initiative Conference, the 2015 American Finance Association

Conference, the 2015 Financial Intermediation Research Society Conference, the 2015 Northern Finance

Conference, the 2015 European Finance Association Conference, and the 2016 Finance Down Under

Conference for helpful comments and discussions. We are grateful for funding from the Europlace Institute

of Finance, the Paul Woolley Centre at the London School of Economics, and the QUANTVALLEY/FdR:

Quantitative Management Initiative.

The Booms and Busts of Beta Arbitrage

Abstract

Low-beta stocks deliver high average returns and low risk relative to high-beta

stocks, an opportunity for professional investors to “arbitrage” away. We argue

that beta-arbitrage activity instead generates booms and busts in the strategy’s abnormal trading profits. In times of low arbitrage activity, the beta-arbitrage

strategy exhibits delayed correction, taking up to three years for abnormal returns to be realized. In stark contrast, when activity is high, prices overshoot as short-

run abnormal returns are much larger and then revert in the long run. We

document a novel positive-feedback channel operating through firm-level leverage that facilitates these boom and bust cycles. These cyclical patterns also show up

in hedge fund exposures to beta arbitrage, particularly exposures of smaller and thus more nimble funds.

1

I. Introduction

The trade-off of risk and return is a key concept in modern finance. The simplest and

most intuitive measure of risk is market beta – the slope in the regression of a security’s

return on the market return. In the Capital Asset Pricing Model (CAPM) of Sharpe (1964)

and Lintner (1965), market beta is the only risk needed to explain expected returns. More

specifically, the CAPM predicts that the relation between expected return and beta – the

security market line – has an intercept equal to the risk-free rate and a slope equal to the

equity premium.

However, empirical evidence indicates that the security market line is too flat on

average (Black 1972, Frazzini and Pedersen, 2014) and especially so during times of high

expected inflation (Cohen, Polk, and Vuolteenaho 2005), disagreement (Hong and Sraer

2016), and market sentiment (Antoniou, Doukas, and Subrahmanyam 2015). These

patterns are not explained by other well-known asset pricing anomalies such as size, value,

and price momentum.

We study the response of arbitrageurs to this failure of the Sharpe-Lintner CAPM

in order to identify booms and busts of beta arbitrage.1 In particular, we exploit the novel

measure of arbitrage activity introduced by Lou and Polk (2014). They argue that

traditional measures of such activity are flawed, poorly measuring a portion of the inputs

to the arbitrage process, for a subset of arbitrageurs. Lou and Polk’s innovation is to

measure the outcome of the arbitrage process, namely, the correlated price impacts that

can result in excess return comovement in the spirit of Barberis and Shleifer (2003).2

We first confirm that our measure of the excess return comovement, relative to a

benchmark asset pricing model, of beta-arbitrage stocks (labelled 𝐶𝑜𝐵𝐴𝑅) is correlated

1 Schwert (2003) summarizes the performance of many trading strategies and motivated by his results,

argues that practitioners who trade on anomalies after their publication can cause the effects to disappear.

McLean and Pontiff (2016) systematically study the out-of-sample and post-publication performance for 97

variables shown to predict stock returns.

2 See, for example, Barberis, Shleifer and Wurgler (2005), Greenwood and Thesmar (2011), Lou (2012) and

Anton and Polk (2014).

2

with existing measures of arbitrage activity. In particular, we find that time variation in

the level of institutional holdings in low-beta stocks (i.e., stocks in the long leg of the beta

strategy), the assets under management of long-short equity hedge funds, aggregate

liquidity, and the past performance of a typical beta-arbitrage strategy together forecast

roughly 35% of the time-series variation in 𝐶𝑜𝐵𝐴𝑅. These findings suggest that not only

is our measure consistent with existing proxies for arbitrage activity but also that no one

single existing proxy is sufficient for capturing time-series variation in arbitrage activity.

Indeed, one could argue that perhaps much of the unexplained variation in 𝐶𝑜𝐵𝐴𝑅

represents variation in arbitrage activity missed by existing measures.

After validating our measure in this way, we then forecast the cumulative abnormal

returns to beta arbitrage. We first find that when arbitrage activity is relatively high (as

identified by the 20% of the sample period with the highest values of 𝐶𝑜𝐵𝐴𝑅), abnormal

returns to beta-arbitrage strategies occur relatively quickly, within the first six months of

the trade. In contrast, when arbitrage activity is relatively low (as identified by the 20%

of the sample period with the lowest values of 𝐶𝑜𝐵𝐴𝑅), any four-factor abnormal returns

to beta-arbitrage strategies take much longer to materialize, appearing three years after

putting on the trade.

These effects are both economically and statistically significant. When beta-

arbitrage activity is low, the abnormal four-factor returns on beta arbitrage are

statistically insignificant from zero in the two years after portfolio formation. For the

patient arbitrageur, in year 3, the strategy earns abnormal four-factor returns of 0.59%

per month with a t-statistic of 2.22. In stark contrast, for those periods when arbitrage

activity is high, the abnormal four-factor returns to beta arbitrage average 1.08% per

month with a t-statistic of 2.63 in the six months after the trade.

We then show that the stronger performance of beta-arbitrage activities during

periods of high beta-arbitrage activity can be linked to subsequent reversal of those profits.

In particular, the year 3 abnormal four-factor returns are -0.93% with an associated t-

3

statistic of -2.66. As a consequence, the long-run reversal of beta-arbitrage returns varies

predictably through time in a striking fashion. The post-formation, year-3 spread in

abnormal four-factor returns across periods of low arbitrage activity, when abnormal

returns are predictably positive, and periods of high arbitrage activity, when abnormal

returns are predictably negative, is -1.52%/month (t-statistic = -3.33) or nearly 20%

cumulative in that year.

Our results reveal interesting patterns in the relation between arbitrage returns

and the arbitrage crowd. When beta-arbitrage activity is low, the returns to beta-arbitrage

strategies exhibit significant delayed correction. In contrast, when beta-arbitrage activity

is high, the returns to beta-arbitrage activities reflect strong over-correction due to

crowded arbitrage trading. These results are consistent with time-varying arbitrage

activity generating booms and busts in beta arbitrage.

We argue that these results are intuitive, as it is difficult to know how much

arbitrage activity is pursuing beta arbitrage, and, in particular, the strategy is susceptible

to positive-feedback trading. Specifically, successful bets on (against) low-beta (high-beta)

stocks result in prices for those securities rising (falling). If the underlying firms are

leveraged, this change in price will, all else equal, result in the security’s beta falling

(increasing) further. Thus, not only do arbitrageurs not know when to stop trading the

low-beta strategy, their (collective) trades strengthen the signal. Consequently, beta

arbitrageurs may increase their bets precisely when trading is more crowded.3

Consistent with our novel positive-feedback mechanism, we show that the cross-

sectional spread in betas increases when beta-arbitrage activity is high and particularly

so when beta-arbitrage stocks are relatively more levered. As a consequence, stocks remain

in the extreme beta portfolios for a longer period of time. Our novel positive feedback

channel also has implications for cross-sectional heterogeneity in abnormal returns; we

3 Note that crowded trading may or may not be profitable, depending on how long the arbitrageur holds

the position and how long it takes for any subsequent correction to occur.

4

find that our boom and bust beta-arbitrage cycles are particularly strong among high-

leverage stocks.

A variety of robustness tests confirm our main findings. In particular, we show

that controlling for other factors when either measuring 𝐶𝑜𝐵𝐴𝑅 or when predicting beta-

arbitrage returns does not alter our primary conclusions a) that the excess comovement

of beta-arbitrage stocks forecasts time-varying reversal to beta-arbitrage bets and b) that

the beta spread varies with 𝐶𝑜𝐵𝐴𝑅.

Our findings can also be seen by estimating time variation in the short-run (months

1-6) and long-run (year 3) security market lines, conditioning on 𝐶𝑜𝐵𝐴𝑅. Thus, the

patterns we find are not just due to extreme-beta stocks, but reflect dynamic movements

throughout the entire cross section. In particular, we find that during periods of high beta-

arbitrage activity, the short-term security market line strongly slopes downward,

indicating strong profits to the low-beta strategy, consistent with arbitrageurs expediting

the correction of market misevaluation. However, this correction is excessive, as the long-

run security market line dramatically slopes upwards. In contrast, during periods of low

beta-arbitrage activity, the short-term security market line is weakly upward sloping.

During these low-arbitrage periods, we do not find any downward slope to the security

market line until the long-run.

A particularly compelling robustness test involves separating 𝐶𝑜𝐵𝐴𝑅 into excess

comovement among low-beta stocks occurring when these stocks have relatively high

returns (i.e., capital flowing into low beta stocks and pushing up the prices) vs. excess

comovement occurring when low-beta stocks have relatively low returns—i.e., upside

versus downside comovement. Under our interpretation of the key findings, it is the former

that should track time-series variation in expected beta-arbitrage returns, as that

particular direction of comovement is consistent with trading aiming to correct the beta

anomaly. Though estimates of upside 𝐶𝑜𝐵𝐴𝑅 are naturally much noisier, our evidence

confirms the intuition above: our main results are primarily driven by upside 𝐶𝑜𝐵𝐴𝑅.

5

Finally, Shleifer and Vishny (1997) link the extent of arbitrage activity to limits

to arbitrage. Based on their logic, trading strategies that bet on firms that are cheaper to

arbitrage (e.g., larger stocks, more liquid stocks, or stocks with lower idiosyncratic risk)

should have more arbitrage activity. This idea of limits to arbitrage motivates tests

examining cross-sectional heterogeneity in our findings. We show that our results

primarily occur in those stocks that provide the least limits to arbitrage: large stocks,

liquid stocks, and stocks with low idiosyncratic volatility. This cross-sectional

heterogeneity in the effect is again consistent with the interpretation that arbitrage

activity causes much of the time-varying patterns we document.

With these patterns in hand, we take a closer look at how our measure reflects

investment activity in this strategy in the cross-section of mutual funds and hedge funds.

Specifically, we show that the typical long-short equity hedge fund increases its beta-

arbitrage exposure when 𝐶𝑜𝐵𝐴𝑅 is relatively high (that is, when short-term beta arbitrage

returns are also higher). However, the ability of hedge funds to time the strong

overreaction that occurs when 𝐶𝑜𝐵𝐴𝑅 is high declines with assets under management.

During booms in beta-arbitrage, small hedge funds have positive exposures to a low-beta

factor that are more than twice as big as their large fund counterparts. In stark contrast,

mutual funds have an insignificant negative exposure to low-beta strategies (i.e., they

overweight high-beta stocks), which does not vary with 𝐶𝑜𝐵𝐴𝑅.

The organization of our paper is as follows. Section II summarizes the related

literature. Section III describes the data and empirical methodology. We detail our

empirical findings regarding beta-arbitrage activity and predictable patterns in returns in

section IV, and present key tests of our economic mechanism in Section V. Section VI

concludes.

II. Related Literature

Our results shed new light on the risk-return trade-off, a cornerstone of modern asset

pricing research. This trade-off was first established in the Sharpe-Lintner CAPM, which

6

argues that the market portfolio is mean-variance efficient. Consequently, a stock’s

expected return is a linear function of its market beta, with a slope equal to the equity

premium and an intercept equal to the risk-free rate.

However, mounting empirical evidence is inconsistent with the CAPM. Black,

Jensen, and Scholes (1972) were the first to show carefully that the security market line

is too flat on average. Put differently, the risk-adjusted returns of high beta stocks are

too low relative to those of low-beta stocks. This finding was subsequently confirmed in

an influential study by Fama and French (1992). Blitz and van Vliet (2007) and Baker,

Bradley, and Taliaferro (2014), Frazzini and Pedersen (2014), and Blitz, Pang, and van

Vliet (2013) document that the low-beta anomaly is also present in both non-US developed

markets as well as emerging markets.

Of course, the flat security market line is not the only failing of the CAPM (see

Fama and French 1992, 1993, and 1996). Nevertheless, since this particular issue is so

striking, a variety of explanations have been offered to explain the low-beta phenomenon.

Black (1972) and more recently Frazzini and Pedersen (2014) argue that leverage-

constrained investors, such as mutual funds, tend to deviate from the capital market line

and invest in high beta stocks to pursue higher expected returns, thus causing these stocks

to be overpriced relative to the CAPM benchmark.4,5,6

Cohen, Polk, and Vuolteenaho (2005) derive the cross-sectional implications of the

CAPM in conjunction with the money illusion story of Modigliani and Cohn (1979). They

show that money illusion implies that, when inflation is low or negative, the compensation

for one unit of beta among stocks is larger (and the security market line steeper) than the

rationally expected equity premium. Conversely, when inflation is high, the compensation

4 Jylhä (2017) provides evidence in support of this interpretation using Federal Reserve changes in initial

margin requirements.

5 See also Karceski (2002), Baker, Bradley, and Wurgler (2011) and Buffa, Vayanos, and Woolley (2014)

for related explanations based on benchmarking of institutional investors.

6 Koijen and Yogo (2017) provide a general framework for modeling the role of institutions in asset markets.

7

for one unit of beta among stocks is lower (and the security market line shallower) than

what the overall pricing of stocks relative to bills would suggest. Cohen, Polk, and

Vuolteenaho provide empirical evidence in support of their theory.

Hong and Sraer (2016) provide an alternative explanation based on Miller’s (1977)

insights. In particular, they argue that investors disagree about the value of the market

portfolio. This disagreement, coupled with short sales constraints, can lead to

overvaluation, and particularly so for high-beta stocks, as these stocks allow optimistic

investors to tilt towards the market. Further, Kumar (2009) and Bali, Cakici, and

Whitelaw (2011) show that high risk stocks can indeed underperform low risk stocks, if

some investors prefer volatile, skewed returns, in the spirit of the cumulative prospect

theory as modeled by Barberis and Huang (2008). Building on arguments in Stambaugh,

Yu, and Yuan (2015), Liu, Stambaugh, and Yuan (2017) attribute the beta anomaly to

the positive correlation between market beta and idiosyncratic volatilities.7

A natural question is why sophisticated investors, who can lever up and sell short

securities at relatively low costs, do not fully take advantage of this anomaly and thus

restore the theoretical relation between risk and returns. Our paper is aimed at addressing

this exact question. Our premise is that professional investors indeed take advantage of

this low-beta return pattern, often in dedicated strategies that buy low-beta stocks and

sell high-beta stocks. However, the amount of capital that is dedicated to this low-beta

strategy is both time varying and unpredictable from arbitrageurs’ perspectives, thus

resulting in periods where the security market line remains too flat—i.e., too little

arbitrage capital, as well as periods where the security market line becomes overly steep—

i.e., too much arbitrage capital.

Not all arbitrage strategies have these issues. Indeed, some strategies have a natural

anchor that is relatively easily observed (Stein 2009). For example, it is straightforward

7 In addition, Campbell, Giglio, Polk, and Turley (2018) document that high-beta stocks hedge time-

variation in the aggregate market’s return volatility, offering a potential neoclassical explanation for the

low-beta anomaly.

8

to observe the extent to which an ADR is trading at a price premium (discount) relative

to its local share. This ADR premium/discount is a clear signal to an arbitrageur of an

opportunity and, in fact, arbitrage activity keeps any price differential small with

deviations disappearing within minutes.8 Importantly, if an unexpectedly large number of

ADR arbitrageurs pursue a particular trade, the price differential narrows. An individual

ADR arbitrageur can then adjust his or her demand accordingly.

There is, however, no easy anchor for beta arbitrage.9 Further, we argue that the

difficulty in identifying the amount of beta-arbitrage capital is exacerbated by a novel,

indirect positive-feedback channel.10 Namely, beta-arbitrage trading can lead to the cross-

sectional beta spread increasing when firms are levered. As a consequence, stocks in the

extreme beta deciles are more likely to remain in these extreme groups, with more extreme

beta values, when arbitrage trading becomes excessive. Given that beta arbitrageurs rely

on realized beta as their trading signal, this beta expansion resulting from leverage

effectively causes a positive feedback loop in the beta-arbitrage strategy.

III. Data and Methodology

The main dataset used in this study is the stock return data from the Center for Research

in Security Prices (CRSP). Following prior studies on the beta-arbitrage strategy, we

include in our study all common stocks on NYSE, Amex, and NASDAQ. We then

8 Rösch (2014) studies various properties of ADR arbitrage. For his sample of 72 ADR home stock pairs,

the average time it takes until a ADR/home stock price deviation disappears is 252 seconds. For an

institutional overview of this strategy, see J.P. Morgan (2014).

9 Polk, Thompson, and Vuolteenaho (2006) use the Sharpe-Lintner CAPM to relate the cross-sectional beta

premium to the equity premium. They show how the divergence of the two types of equity-premium

measures implies a time-varying trading opportunity for beta arbitrage. Their methods are quite

sophisticated and produce signals about the time-varying attractiveness of beta-arbitrage that, though useful

in predicting beta-arbitrage returns, are still, of course, quite noisy.

10 The idea that positive-feedback strategies are prone to destabilizing behaviour goes back to at least

DeLong, Shleifer, Summers, and Waldmann (1990). In contrast, negative-feedback strategies like ADR

arbitrage or value investing are less susceptible to destabilizing behaviour by arbitrageurs, as the price

mechanism mediates any potential congestion. See Stein (2009) for a discussion of these issues.

9

augment the stock return data with institutional ownership in individual stocks provided

by Thompson Financial. We further obtain information on assets under management of

long-short equity hedge funds from Lipper’s Trading Advisor Selection System (TASS).

Since the assets managed by hedge funds grow substantially in our sample period, we

detrend this variable. In addition, we use fund-level data on hedge fund returns and AUM.

We also construct, as controls, a list of variables that have been shown to predict

future beta-arbitrage strategy returns. Specifically, a) following Cohen, Polk, and

Vuolteenaho (2005), we construct a proxy for expected inflation using an exponentially

weighted moving average (with a half-life of 36 months) of past log growth rates of the

producer-price index; b) we also include in our study the sentiment index proposed by

Baker and Wurgler (2006, 2007); c) following Hong and Sraer (2016), we construct an

aggregate disagreement proxy as the beta-weighted standard deviation of analysts’ long-

term growth rate forecasts; d) finally, following Frazzini and Pedersen (2014), we use the

Ted spread—the difference between the LIBOR rate and the US Treasury bill rate—as a

measure of financial intermediaries’ funding constraints.

We begin our analysis in January 1970 (i.e., our first measure of beta arbitrage

crowdedness is computed as of December 1969), as that was when the low-beta anomaly

was first recognized by academics.11 At the end of each month, we sort all stocks into

deciles (in some cases vigintiles) based on their pre-ranking market betas. Following prior

literature, we calculate pre-ranking betas using daily returns in the past twelve months

(with at least 200 daily observations). Our results are similar if we use monthly returns,

or different pre-ranking periods. To account for illiquidity and non-synchronous trading,

we include on the right hand side of the regression equation five lags of the excess market

return, in addition to the contemporaneous excess market return. The pre-ranking beta is

simply the sum of the six coefficients from the OLS regression.

11 Though eventually published in 1972, Black, Jensen, and Scholes had been presented as early as August

of 1969. Mehrling’s (2005) biography of Fischer Black details the early history of the low-beta anomaly.

10

We then compute pairwise partial correlations using 52 (non-missing) weekly

returns for all stocks in each decile in the portfolio ranking period. We control for the

Fama-French three factors when computing these partial correlations to purge out any

comovement in stocks induced by known risk factors. We measure the excess comovement

of stocks involved in beta arbitrage (𝐶𝑜𝐵𝐴𝑅) as the average pairwise partial correlation

in the lowest market beta decile. We focus on the low-beta decile as these stocks tend to

be larger, more liquid, and have lower idiosyncratic volatility compared to the highest-

beta decile; thus, our measurement of excess comovement will be less susceptible to issues

related to asynchronous trading and measurement noise. 12 We operationalize this

calculation by computing the average correlation of the three-factor residual of every stock

in the lowest beta decile with the rest of the stocks in the same decile:

𝐶𝑜𝐵𝐴𝑅 =1

𝑁∑ 𝑝𝑎𝑟𝑡𝑖𝑎𝑙𝐶𝑜𝑟𝑟(𝑟𝑒𝑡𝑟𝑓𝑖

𝐿 , 𝑟𝑒𝑡𝑟𝑓−𝑖𝐿 |𝑚𝑘𝑡𝑟𝑓, 𝑠𝑚𝑏, ℎ𝑚𝑙)

𝑁

𝑖=1

,

where 𝑟𝑒𝑡𝑟𝑓𝑖𝐿 is the weekly return of stock 𝑖 in the (L)owest beta decile, 𝑟𝑒𝑡𝑟𝑓−𝑖

𝐿 is the

weekly return of the equal-weight lowest beta decile excluding stock 𝑖 , and 𝑁 is the

number of stocks in the lowest beta decile. We have also measured 𝐶𝑜𝐵𝐴𝑅 using returns

that are orthogonalized not only to the Fama-French factors but also to each stock’s

industry return or to other empirical priced factors, and our conclusions continue to hold.

We present these and many other robustness tests in Table IV.

In the following period, we then form a zero-cost portfolio that goes long the value-

weight portfolio of stocks in the lowest market beta decile and short the value-weight

portfolio of stocks in the highest market beta decile. We track the cumulative abnormal

returns of this zero-cost long-short portfolio in months 1 through 36 after portfolio

formation. To summarize the timing of our empirical exercise, year 0 is our portfolio

formation year (during which we also measure 𝐶𝑜𝐵𝐴𝑅), year 1 is the holding year, and

12 Our results are robust to measuring 𝐶𝑜𝐵𝐴𝑅 using a pooled sample of high- and low-beta stocks (putting

a negative sign in front of the returns of high-beta stocks), as well as the (minus) cross-correlation between

high- and low-beta deciles.

11

years 2 and 3 are our post-holding period, to detect any (conditional) long-run reversal to

the beta-arbitrage strategy.

IV. Main Results

We first document simple characteristics of our arbitrage activity measure. Table I Panel

A indicates that there is significant excess correlation among low-beta stocks on average

and that this pairwise correlation varies substantially through time; specifically, the mean

of 𝐶𝑜𝐵𝐴𝑅 is 0.11 varying from a low of 0.04 to a high of 0.20.

Panel B of Table I examines 𝐶𝑜𝐵𝐴𝑅’s correlation with existing measures linked to

time variation in the expected abnormal returns to beta-arbitrage strategies. We find that

𝐶𝑜𝐵𝐴𝑅 is high when disagreement is high, with a correlation of 0.34. 𝐶𝑜𝐵𝐴𝑅 is also

positively correlated with the Ted spread, consistent with a time-varying version of Black

(1972), though the Ted spread does not forecast (or in some cases forecasts in the wrong

direction) time variation in expected abnormal returns to beta-arbitrage strategies

(Frazzini and Pederson 2014). 𝐶𝑜𝐵𝐴𝑅 is negatively correlated with the expected inflation

measure of Cohen, Polk, and Vuolteenaho. However, in results not shown, the correlation

between expected inflation and 𝐶𝑜𝐵𝐴𝑅 becomes positive for the subsample from 1990-

2010, consistent with arbitrage activity eventually taking advantage of this particular

source of time-variation in beta-arbitrage profits. There is little to no correlation between

𝐶𝑜𝐵𝐴𝑅 and sentiment.

Figure 1 plots 𝐶𝑜𝐵𝐴𝑅 as of the end of each December. Note that we do not

necessarily expect a trend in this measure. Though there is clearly more capital invested

in beta-arbitrage strategies, in general, markets are also deeper and more liquid.

Nevertheless, after an initial spike in December 1971, 𝐶𝑜𝐵𝐴𝑅 trends slightly upward for

the rest of the sample. However, there are clear cycles around this trend. These cycles

tend to peak before broad market declines. Also, note that 𝐶𝑜𝐵𝐴𝑅 is essentially

12

uncorrelated with market volatility. A regression of 𝐶𝑜𝐵𝐴𝑅 on contemporaneous realized

market volatility produces a loading of -0.23 with a t-statistic of -0.79.

Consistent with our measure tracking arbitrage activity, Appendix Table A1 shows

that 𝐶𝑜𝐵𝐴𝑅 is persistent in event time. Specifically, the correlation between 𝐶𝑜𝐵𝐴𝑅

measured in year 0 and year 1 for the same set of stocks is 0.15. In fact, year-0 𝐶𝑜𝐵𝐴𝑅

remains highly correlated with subsequent values of 𝐶𝑜𝐵𝐴𝑅 for the same stocks all the

way out to year 3. The average value of 𝐶𝑜𝐵𝐴𝑅 remains high as well. Recall that in year

0, the average excess correlation is 0.11. We find that in years 1, 2, and 3, the average

excess correlation of these same stocks remains around 0.07.13

IV.A. Determinants of 𝐶𝑜𝐵𝐴𝑅

To confirm that our measure of beta-arbitrage is sensible, we estimate regressions

forecasting 𝐶𝑜𝐵𝐴𝑅 with four variables that are often used to proxy for arbitrage activity.

The first variable we use is the aggregate institutional ownership (Inst Own) of the low-

beta decile—i.e., stocks in the long leg of the beta strategy—based on 13F filings. We

include institutional ownership as these investors are typically considered smart money,

at least relative to individuals, and we focus on their holdings in the low-beta decile as

we do not observe their short positions in the high-beta decile. We also include the assets

under management (AUM) of long-short equity hedge funds, the prototypical arbitrageur.

Finally, we include a measure of the past profitability of beta-arbitrage strategies, the

realized four-factor alpha of Frazzini and Pedersen’s BAB factor. Intuitively, more

arbitrageurs should be trading the low-beta strategy after the strategy has performed well

in recent past.

All else equal, we expect 𝐶𝑜𝐵𝐴𝑅 to be lower if markets are more liquid. However,

as arbitrage activity is endogenous, times when markets are more liquid may also be times

13 𝐶𝑜𝐵𝐴𝑅 is essentially uncorrelated with a similar measure of excess comovement based on the fifth and

sixth beta deciles.

13

when arbitrageurs are more active. Indeed, Cao, Chen, Liang, and Lo (2013) show that

hedge funds increase their activity in response to increases in aggregate liquidity.

Following Cao, Chen, Liang, and Lo, we further include past market liquidity as proxied

by the Pastor and Stambaugh (2003) liquidity factor (PS liquidity) in our regressions to

measure which channel dominates.

All regressions in Table II include a trend to ensure that our results are not

spurious. We also report a second specification that includes variables that arguably

should forecast beta-arbitrage returns: the inflation, sentiment, and disagreement indices

as well as the Ted spread. We measure these variables contemporaneously with 𝐶𝑜𝐵𝐴𝑅

as we will be running horse races against these variables in our subsequent analysis.

Regression (1) in Table II documents that Inst Own, AUM, and PS liquidity forecast

𝐶𝑜𝐵𝐴𝑅, with an R2 of approximately 35%.14 Regression (2) shows that two of the extant

predictors of beta-arbitrage returns help explain 𝐶𝑜𝐵𝐴𝑅. The Ted spread, adds some

incremental explanatory power, with the sign of the coefficient consistent with

arbitrageurs taking advantage of potential time-variation in beta-arbitrage returns linked

to this channel. Indeed, as we show later, the Ted spread does a poor job forecasting beta-

arbitrage returns in practice, perhaps because arbitrageurs have compensated

appropriately for this potential departure from Sharpe-Lintner pricing. The disagreement

measure also helps explain variation in 𝐶𝑜𝐵𝐴𝑅. Finally, in this full specification, past

profitability of a prototypical beta-arbitrage strategy strongly forecasts relatively high

arbitrage activity going forward. It seems reasonable that strong past performance of an

investment strategy may result in the strategy becoming more popular.

Overall, these findings make us comfortable in our interpretation that 𝐶𝑜𝐵𝐴𝑅 is

related to arbitrage activity and distinct from existing measures of opportunities in beta

14 We choose to forecast 𝐶𝑜𝐵𝐴𝑅 in a predictive regression rather than explain 𝐶𝑜𝐵𝐴𝑅 in a contemporaneous

regression simply to reduce the chance of a spurious fit. However, our results are robust to estimating

contemporaneous versions of these regressions.

14

arbitrage. As a consequence, we turn to the main analysis of the paper, the short- and

long-run performance of beta-arbitrage returns, conditional on 𝐶𝑜𝐵𝐴𝑅.

IV.B. Forecasting Beta-Arbitrage Returns

Table III forecasts the abnormal returns on the standard beta-arbitrage strategy as a

function of investment horizon, conditional on 𝐶𝑜𝐵𝐴𝑅. Panel A examines Fama and

French (1993) three-factor-adjusted returns while Panel B studies abnormal returns

relative to the four-factor model of Carhart (1997).15 In each panel, we measure the

average abnormal returns in the first six months subsequent to the beta-arbitrage trade,

months seven through 12, and then those occurring in years two and three. We also report

the average abnormal returns across years two and three combined. These returns are

measured as a function of the value of 𝐶𝑜𝐵𝐴𝑅 as of the end of the beta formation period.

In particular, we split the sample into five 𝐶𝑜𝐵𝐴𝑅 quintiles.

We focus on the four-factor estimates. Pursuing beta arbitrage when arbitrage

activity is low takes patience. Abnormal four-factor returns are statistically insignificant

in the first year for the bottom four 𝐶𝑜𝐵𝐴𝑅 groups.16 Abnormal returns only become

statistically significant for the lowest 𝐶𝑜𝐵𝐴𝑅 group in the third year. In the lowest 𝐶𝑜𝐵𝐴𝑅

period, the four-factor alpha is 0.59%/month with an associated t-statistic of 2.22.17

However, as beta-arbitrage activity increases, the abnormal returns arrive sooner

and stronger. For the highest 𝐶𝑜𝐵𝐴𝑅 group, the abnormal four-factor returns average

15 We compute the four-factor alpha for each event month after portfolio formation by taking the average

across all the calendar months in our sample.

16 𝐶𝑜𝐵𝐴𝑅 has a non-linear, U-shaped relation with beta arbitrage returns in the following year. As can be

seen in Appendix Tables A2 and A3, once we include both 𝐶𝑜𝐵𝐴𝑅 and 𝐶𝑜𝐵𝐴𝑅2 in the regression to forecast

future beta arbitrage returns, both coefficients are statistically and economically significant, but have

opposite signs. In other words, beta arbitrage returns are relatively high following periods of both extreme

high arbitrage activity (when arbitrageurs collectively bet against mispricing) and extreme low arbitrage

activity (when great opportunities are left unexploited).

17 We have also separately examined the long and short legs of beta arbitrage (i.e., low-beta vs. high-beta

stocks). Around 40% of our return effect comes from the long leg, and the remaining 60% from the short

leg.

15

1.08%/month in the six months immediately subsequent to the beta-arbitrage trade. This

finding is statistically significant with a t-statistic of 2.63, though the difference between

abnormal returns in high and low 𝐶𝑜𝐵𝐴𝑅 periods, despite being economically large at

0.75%/month, is statistically insignificant. (In a regression context as in table V, where

we have more power to isolate the effect of crowded trading, 𝐶𝑜𝐵𝐴𝑅 positively forecasts

beta arbitrage returns in the first six months.)

The key finding of our paper is that these quicker and stronger beta-arbitrage

returns can be linked to subsequent reversal in the long run. Specifically, in year three,

the abnormal four-factor return to beta arbitrage when 𝐶𝑜𝐵𝐴𝑅 is high is -0.93%/month,

with a t-statistic of -2.66. These abnormal returns are dramatically different from their

corresponding values when 𝐶𝑜𝐵𝐴𝑅 is low; the difference in year 3 abnormal four-factor

returns is -1.52%/month (t-statistic = -3.33).18

Since splitting the long run at year 2 is arbitrary, Table III also reports the results

combining years 2 and 3 together. The patterns remain, and these estimates are

monotonically decreasing in 𝐶𝑜𝐵𝐴𝑅.

Figure 2 summarizes these patterns by plotting the cumulative abnormal four-

factor returns to beta arbitrage during periods of high and low 𝐶𝑜𝐵𝐴𝑅, accumulating

abnormal returns up to 60 months post portfolio formation. We include in the plot the

cumulative four-factor returns during median 𝐶𝑜𝐵𝐴𝑅 periods as well. This figure clearly

shows that there is a significant delay in abnormal trading profits to beta arbitrage when

beta-arbitrage activity is low. However, when beta-arbitrage activity is high, beta

arbitrage results in price overshooting, as evidenced by the initial price run-up and

subsequent reversal that we document. We argue that trading of the low-beta anomaly is

initially stabilizing, then, as the trade becomes crowded, turns destabilizing, causing prices

to overshoot. The bottom panel of Figure 2 further shows that the run-up to the beta

18 We postpone the discussion of conditional abnormal returns to beta arbitrage (as shown in the last row

of Table II) to Section V.C.

16

arbitrage strategy during high 𝐶𝑜𝐵𝐴𝑅 periods starts in the formation period, consistent

with the view that arbitrageurs may use shorter windows to calculate beta. If this

formation-period runup is included in the accumulation of abnormal return, we can

account for effectively all of the reversal.

IV.C. Robustness of Key Results

Table IV examines variations to our methodology to ensure that our finding of time-

varying reversal of beta-arbitrage profits is robust. For simplicity, we report the difference

in returns to the beta strategy between the high and low 𝐶𝑜𝐵𝐴𝑅 groups in two versions

of the long run (year 3 and years 2-3). For reference, the first row of Table IV reports the

baseline results from Table III Panel B.

In row two, we conduct the same analysis for the sub-period before our sample

(1927-1969). Of course, this sample not only predates the discovery of the low-beta

anomaly but also is a period where there is much less arbitrage activity in general, at

least explicitly organized as such. Thus, this period could be thought of as a placebo test

of our story. Consistent with our paper’s explanation, we find no statistically significant

link between 𝐶𝑜𝐵𝐴𝑅 and the reversal of beta-arbitrage returns.

Our remaining subsample analysis excludes potential outlier years. We find that

our results remain robust if we exclude the tech bubble crash (2001) or the recent financial

crisis (2007-2009) from our sample.

In rows five through eight, we report the results from similar tests using extant

variables linked to potential time variation in beta-arbitrage profits. None of the four

variables are associated with time variation in long-run abnormal returns. Thus, our

𝐶𝑜𝐵𝐴𝑅 measure has not simply repackaged an effect linked to existing forecasting

variables.

In rows nine through 15, we consider alternative definitions of 𝐶𝑜𝐵𝐴𝑅. In the ninth

row, we control for UMD when computing 𝐶𝑜𝐵𝐴𝑅. In row 10, we separate HML into its

17

large cap and small cap components. In Row 11, we report results based on Fama and

French (2015a, 2015b) five-factor returns. In Row 12, we perform the entire analysis on

an industry-adjusted basis by sorting stocks into beta deciles within industries. Row 13

uses the correlation between the high-beta and low-beta deciles as a measure of arbitrage

activity, with lower values indicating more activity.19

Rows 14 and 15 split 𝐶𝑜𝐵𝐴𝑅 into upside and downside components. Specifically,

we measure the following

𝐶𝑜𝐵𝐴𝑅𝑈 =1

𝑁∑ 𝑝𝑎𝑟𝑡𝑖𝑎𝑙𝐶𝑜𝑟𝑟(𝑟𝑒𝑡𝑟𝑓𝑖

𝐿, 𝑟𝑒𝑡𝑟𝑓−𝑖𝐿 |𝑚𝑘𝑡𝑟𝑓, 𝑠𝑚𝑏, ℎ𝑚𝑙, 𝑟𝑒𝑡𝑟𝑓𝐿 > 𝑚𝑒𝑑𝑖𝑎𝑛(𝑟𝑒𝑡𝑟𝑓𝐿))

𝑁

𝑖=1

𝐶𝑜𝐵𝐴𝑅𝐷 =1

𝑁∑ 𝑝𝑎𝑟𝑡𝑖𝑎𝑙𝐶𝑜𝑟𝑟(𝑟𝑒𝑡𝑟𝑓𝑖

𝐿, 𝑟𝑒𝑡𝑟𝑓−𝑖𝐿 |𝑚𝑘𝑡𝑟𝑓, 𝑠𝑚𝑏, ℎ𝑚𝑙, 𝑟𝑒𝑡𝑟𝑓𝐿 < 𝑚𝑒𝑑𝑖𝑎𝑛(𝑟𝑒𝑡𝑟𝑓𝐿))

𝑁

𝑖=1

Separating 𝐶𝑜𝐵𝐴𝑅 in this way allows us to distinguish between excess comovement tied

to strategies buying low-beta stocks (such as those followed by beta arbitrageurs) and

strategies selling low-beta stocks (such as leveraged-constrained investors modeled by

Black (1972)). Consistent with our interpretation, we find that only 𝐶𝑜𝐵𝐴𝑅𝑈 forecasts

time variation in the short- and long-run expected returns to beta arbitrage (whereas

𝐶𝑜𝐵𝐴𝑅𝐷 does not).

Rows 16-22 document that our results are robust to replacing 𝐶𝑜𝐵𝐴𝑅 with

residual 𝐶𝑜𝐵𝐴𝑅. In particular, we orthogonalize 𝐶𝑜𝐵𝐴𝑅 to measures of arbitrage activity

in momentum and value (Lou and Polk 2014), the average correlation in the market

(Pollet and Wilson 2010), the past volatility of beta-arbitrage returns, the volatility of

market returns over the twelve-month period corresponding to the measurement of

𝐶𝑜𝐵𝐴𝑅, a trend, and lagged 𝐶𝑜𝐵𝐴𝑅 (the year -1 average pairwise excess correlation of

low-beta stocks identified in year 0). Finally, in row 23, we measure abnormal returns

19 We also compute 𝐶𝑜𝐵𝐴𝑅 based on a pooled sample of high- and low-beta stocks, putting a negative sign

in front of the returns of high-beta stocks. The results are similar. For example, after controlling for other

confounding factors, the difference in three-factor alpha of beta arbitrage in year three after portfolio

formation between high and low 𝐶𝑜𝐵𝐴𝑅 periods is -0.96%/month (t-statistic = -2.24), and that same

difference in four-factor alpha is -0.77%/month (t-statistic = -1.79).

18

using a six-factor model that augments the Fama-French five-factor model with

momentum.

In all cases, 𝐶𝑜𝐵𝐴𝑅 continues to predict time-variation in year 3 returns. The

estimates are always economically significant, with most point estimates larger than

1%/month. Statistical significance is always strong as well, with most t-statistic larger

than 3. Taken together, these results confirm that our measure of crowded beta arbitrage

robustly forecasts times of strong reversal to beta-arbitrage strategies.

In Table V, we report the results of regressions forecasting the abnormal four-factor

returns to beta-arbitrage spread bets, while controlling for other predictors of beta-

arbitrage returns. Unlike Table II, these regressions exploit not just the ordinal but also

the cardinal aspect of 𝐶𝑜𝐵𝐴𝑅. Moreover, these regressions not only confirm that our

findings are robust to existing measures of the profitability of beta arbitrage, they also

document the relative extent to which existing measures forecast abnormal returns to

beta-arbitrage strategies in the presence of 𝐶𝑜𝐵𝐴𝑅.

Regressions (1)-(3) in Table V forecast time-series variation in abnormal beta-

arbitrage returns in months 1-6. Regression (1) confirms that 𝐶𝑜𝐵𝐴𝑅 strongly forecasts

beta-arbitrage four-factor alphas over the full sample. Regression (2) then includes

controls that are available over the entire sample. These include the inflation and

sentiment indices, market volatility, and a version of Cohen, Polk, and Vuolteenaho’s

(2003) value spread for the beta deciles in question. 𝐶𝑜𝐵𝐴𝑅 continues to reliably describe

time-variation in abnormal four-factor returns on the low-beta-minus-high-beta strategy,

with only the sentiment index providing any additional explanatory power. Over the

shorter sample period where both aggregate disagreement and the Ted spread are

available, 𝐶𝑜𝐵𝐴𝑅 retains its economic significance, but becomes marginally statistically

significant in forecasting time-variation in the abnormal returns to the beta-arbitrage

strategy.

19

Regressions (4)-(6) of Table V forecast the returns on beta-arbitrage strategies in

year 3. The message from these regressions concerning the main result of the paper is

clear; 𝐶𝑜𝐵𝐴𝑅 strongly forecasts a time-varying reversal, while none of the extant variables

has any predictive power for long-run buy-and-hold beta-arbitrage returns.

IV.D. Predicting the Security Market Line

Our results can also be seen from time variation in the shape of the security market line

(SML) as a function of lagged 𝐶𝑜𝐵𝐴𝑅. Such an approach can help ensure that the time-

variation we document is not restricted to a small subset of extreme-beta stocks, but

instead is a robust feature of the cross-section. (We argue that beta arbitrage activity can

affect the entire cross-section of stocks rather than just the extreme deciles, because

arbitrageurs may bet against the low-beta anomalies by selecting portfolio weights that

are inversely proportional to the market beta.) At the end of each month, we sort all

stocks into 20 value-weighted portfolios by their pre-ranking betas.20 We track these 20

portfolio returns both in months 1-6 and months 25-36 after portfolio formation to

compute short-term and long-term post-ranking betas, and, in turn, to construct our

short-term and long-term security market lines.

For the months 1-6 portfolio returns, we then compute the post-ranking betas by

regressing each of the 20 portfolios’ value-weighted monthly returns on market excess

returns. Following Fama and French (1992), we use the entire sample to compute post-

ranking betas. That is, we pool together these six monthly returns across all calendar

months to estimate the portfolio beta. We estimate post-ranking betas for months 25-36

in a similar fashion. The two sets of post-ranking betas are then labelled 𝛽11, ..., 𝛽20

1 and

𝛽125, ..., 𝛽20

25.

20 We sort stocks into vigintiles in order to increase the statistical precision of our cross-sectional estimate.

However, Appendix Table A4 confirms that our results are virtually identical if we instead sort stocks into

deciles.

20

To calculate the intercept and slope of the short-term and long-term security

market lines, we estimate the following cross-sectional regressions:

short-term SML: 𝑋𝑅𝑒𝑡𝑖,𝑡1 = 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡

1 + 𝑠𝑙𝑜𝑝𝑒𝑡1𝛽𝑖

1,

long-term SML: 𝑋𝑅𝑒𝑡𝑖,𝑡25 = 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡

25 + 𝑠𝑙𝑜𝑝𝑒𝑡25𝛽𝑖

25,

where 𝑋𝑅𝑒𝑡𝑖,𝑡1 is portfolio 𝑖’s monthly excess returns in months 1 through 6, and 𝑋𝑅𝑒𝑡𝑖,𝑡

25

is portfolio 𝑖’s monthly returns in months 25 through 36. These two regressions then give

us two time-series of coefficient estimates of the intercept and slope of the short-term and

long-term security market lines: ( 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡1, 𝑠𝑙𝑜𝑝𝑒𝑡

1 ) and ( 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡25, 𝑠𝑙𝑜𝑝𝑒𝑡

25 ),

respectively. As the average excess returns and post-ranking betas are always measured

at the same point in time, the pair (𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡1, 𝑠𝑙𝑜𝑝𝑒𝑡

1) fully describes the security market

line in the short run, while (𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑡25, 𝑠𝑙𝑜𝑝𝑒𝑡

25) captures the security market line two

years later.

We then examine how these intercepts and slopes vary as a function of our measure

of beta-arbitrage capital. In particular, we conduct an OLS regression of the intercept and

slope measured in each month on lagged 𝐶𝑜𝐵𝐴𝑅. As can be seen from Table VI, the

intercept of the short-term security market line significantly increases in 𝐶𝑜𝐵𝐴𝑅, and its

slope significantly decreases in 𝐶𝑜𝐵𝐴𝑅. The top panel of Figure 3 shows this pattern

graphically. During high 𝐶𝑜𝐵𝐴𝑅—i.e., high beta-arbitrage capital—periods, the short-

term security market line strongly slopes downward, indicating strong profits to the low-

beta strategy, consistent with arbitrageurs expediting the correction of market

misevaluation. In contrast, during low 𝐶𝑜𝐵𝐴𝑅—i.e., low beta-arbitrage capital—periods,

the short-term security market line is weakly upward sloping and the beta-arbitrage

strategy, as a consequence, unprofitable, consistent with delayed correction of the beta

anomaly.

The pattern is completely reversed for the long-term security market line. The

intercept of the long-term security market line is significantly negatively related to 𝐶𝑜𝐵𝐴𝑅,

whereas its slope is significantly positively related to 𝐶𝑜𝐵𝐴𝑅. As can be seen from the

21

bottom panel of Figure 3, two years after high 𝐶𝑜𝐵𝐴𝑅 periods, the long-term security

market line turns upward sloping; indeed, the slope is so steep (resulting in a negative

intercept) that the beta strategy loses money, consistent with over-correction of the low

beta anomaly by crowded arbitrage trading. In contrast, after low 𝐶𝑜𝐵𝐴𝑅 periods, the

long-term security market line turns downward sloping, reflecting eventual profitability

of the low-beta strategy in the long run.

IV.E. Smarter Beta-Arbitrage Strategies

One way to measure the economic importance of these boom and bust cycles is through

an out-of-sample calendar-time trading strategy. 21 We combine these time-varying

overreaction and subsequent reversal patterns as follows. We first time the standard beta-

arbitrage strategy using current 𝐶𝑜𝐵𝐴𝑅. If 𝐶𝑜𝐵𝐴𝑅 is above the 80th percentile (of its

distribution up to that point), we are long the long-short beta-arbitrage strategy studied

in Table III for the next six months. Otherwise, we short that portfolio over that time

period. We skip the first three years of our sample to compute the initial distribution as

well as show in-sample results in Panel A of Table VII for the sake of comparison.

In addition, if 𝐶𝑜𝐵𝐴𝑅 from three years ago is below the 20th percentile (of its prior

distribution), we are long for the next twelve months the long-short beta-arbitrage

strategy based on beta estimates from three years ago. Otherwise, we short that portfolio,

again for the next twelve months.

This strategy harvests beta-arbitrage profits much more wisely than unconditional

bets against beta. As can be seen from Panel B of Table VII, the four-factor alpha is 51

basis points per month with a t-statistic of 2.62. The six-factor alpha (where we add the

investment and profitability factors of Fama and French 2015a) remains high at 51 basis

points per month (t-statistic of 2.46). Finally, if we also include the BAB factor of Frazzini

21 This calendar-time approach also ensures that our Newey-West standard errors (for example, the standard

errors of Table III) are not misleading about the statistical significance of our findings.

22

and Pedersen (2014) as a seventh factor, the abnormal return increases to 63 basis points

per month with a t-statistic of 3.10. By comparison, the standard value-weight beta-

arbitrage strategy yields a four-factor alpha of -0.05% per month (t-statistic = -0.17) in

our sample period.

We have also estimated conditional regressions where we interact each factor with

𝐶𝑜𝐵𝐴𝑅. The alpha from this regression is significantly larger at 0.87% per month (t-

statistic of 2.66).22

V. Testing the Economic Mechanism

The previous section documents rich cross-sectional and time-series variation in expected

returns linked to our proxy for arbitrage activity and the low-beta anomaly. In this

section, we delve deeper to test specific aspects of the economic mechanism we argue is

behind these patterns. Our interpretation of these patterns makes specific novel

predictions in terms of the role of firm leverage, the limits to arbitrage, and the reaction

of sophisticated investors to these patterns.

V.A. Beta Expansion

Beta arbitrage can be susceptible to positive-feedback trading. Successful bets on (against)

low-beta (high-beta) stocks result in prices of those securities rising (falling). If the

underlying firms are leveraged, this change in price will, all else equal, result in the

security’s beta falling (increasing) further.23 Thus, not only do arbitrageurs not know when

to stop trading the low-beta strategy, their (collective) trades also affect the strength of

22 We construct an “even-smarter” beta arbitrage strategy by further exploiting differences between high-

leverage and low-leverage firms. In particular, we divide all stocks into four quartiles based on their lagged

leverage ratios. We then go long the smart-beta-strategy formed solely with high-leverage stocks and short

the smart-beta strategy solely with low-leverage stocks. This “even-smarter” beta strategy yields a monthly

alpha of 51bp (t-statistic = 2.93) after controlling for the Fama and French (2016) five factors, momentum

factor, BAB factor, as well as the smart-beta portfolio.

23 The idea that, all else equal, changes in leverage drive changes in equity beta is, of course, the key insight

behind Proposition II of Modigiliani and Miller (1958).

23

the signal. Consequently, beta arbitrageurs may increase their bets precisely when trading

becomes crowded and the expected profitability of the strategy has decreased.

We test this prediction in Table VIII. The dependent variable is the spread in betas

across the high and low value-weight beta decile portfolios, denoted 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑, as of

the end of year 1. The independent variables include lagged 𝐶𝑜𝐵𝐴𝑅, the beta-formation-

period value of 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 (computed from the same set of low- and high- beta stocks

as the dependent variable), the average book leverage quintile (𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒) across the high

and low beta decile portfolios, and an interaction between 𝐶𝑜𝐵𝐴𝑅 and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. Note

that since we estimate beta using 52 weeks of stock returns, the two periods of beta

estimation that determine the change in 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 do not overlap. We include a trend

in all regressions, but our results are robust to not including the trend dummies.

Regression (1) in Table VIII shows that when 𝐶𝑜𝐵𝐴𝑅 is relatively high, future

𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 is also high, controlling for lagged 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑. A one-standard-deviation

increase in 𝐶𝑜𝐵𝐴𝑅 forecasts an increase in 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 of roughly 7% (of the average beta

spread). Regression (2) shows that this is particular true when 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 is also high. If

beta-arbitrage bets were to contain the highest book-leverage quintile stocks, a one-

standard deviation increase in 𝐶𝑜𝐵𝐴𝑅 would increase 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 by nearly 10%.

These results are consistent with a positive feedback channel for the beta-arbitrage

strategy that works through firm-level leverage. In terms of the economic magnitude of

this positive feedback loop, we draw a comparison with the price momentum strategy.

The formation-period spread for a standard price momentum bet in the post-1963 period

is around 115%, while the momentum profit in the subsequent year is close to 12% (e.g.,

Lou and Polk, 2014). Put differently, if we attribute price momentum entirely to positive

feedback trading, such trading increases the initial return spread by about 10% (12%

divided by 115%) in the subsequent year, which is similar in magnitude to the positive

feedback channel we document for beta arbitrage. Table VIII Panel B confirms that these

results are robust to the same methodological variations as in Table IV.

24

Table IX turns to firm-level regressions to document the beta expansion our story

predicts. In particular, we estimate panel regressions forecasting beta with lagged 𝐶𝑜𝐵𝐴𝑅.

At the end of each month, all stocks are sorted into deciles based on their market beta

calculated using daily returns in the past 12 months. The dependent variable is

𝑃𝑜𝑠𝑡 𝑅𝑎𝑛𝑘𝑖𝑛𝑔 𝐵𝑒𝑡𝑎, the change in stock beta from year t to t+1 (again, we use non-

overlapping periods). In addition to 𝐶𝑜𝐵𝐴𝑅, we also include 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, the difference

between a stock’s pre-formation beta and the average pre-formation beta in year t.

𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 is the book leverage of the firm, measured in year t. We also include all double

and triple interaction terms of 𝐶𝑜𝐵𝐴𝑅, 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. Other control variables

include the lagged firm size, book-to-market ratio, lagged one-month and one-year stock

return, and the prior-year idiosyncratic volatility. Time-fixed effects are included in

Columns 3 and 4. Note that since 𝐶𝑜𝐵𝐴𝑅 is a time-series variable, it is subsumed by the

time dummies in those regressions.

In all four regressions, stocks with higher Distance have a higher

𝑃𝑜𝑠𝑡 𝑅𝑎𝑛𝑘𝑖𝑛𝑔 𝐵𝑒𝑡𝑎, consistent with betas being persistent. This persistence is higher when

𝐶𝑜𝐵𝐴𝑅 is relatively high. Our main focus is on the triple interaction among 𝐶𝑜𝐵𝐴𝑅,

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. The persistence in a firm’s beta is significantly stronger when

𝐶𝑜𝐵𝐴𝑅 and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 are high. Taken together, these results are consistent with beta-

arbitrage activity causing the cross-sectional spread in betas to expand.

As a natural extension, our positive feedback channel suggests that booms and

busts of beta arbitrage should be especially strong among more highly levered stocks.

Figure 4 reports results where the sample is split based on leverage. Specifically, at the

beginning of the holding period, we sort stocks into four equal groups using book leverage.

For each leverage quartile, we compute the 𝐶𝑜𝐵𝐴𝑅 return spread – i.e., the difference in

four-factor alpha to the beta arbitrage strategy between high and low 𝐶𝑜𝐵𝐴𝑅 periods.

Reported in the figure is the cumulative difference in the 𝐶𝑜𝐵𝐴𝑅 return spread between

the highest and lowest leverage quartiles over the five years after portfolio formation.

25

As can be seen from the figure, the difference in the 𝐶𝑜𝐵𝐴𝑅 return spread rises

substantially in the first twelve months, by nearly 18% (1.46%*12, t-statistic = 2.14). It

then mostly reverses in the subsequent years. For example, the cumulative 𝐶𝑜𝐵𝐴𝑅 return

spread in years 4-5 is roughly -10% (t-statistic = -2.41). This finding is consistent with

our novel positive feedback channel facilitating excessive arbitrage trading activity that

can potentially destabilize prices.

In results not reported, we have also confirmed that leverage splits enhance the

profitability of the calendar-time strategies studied in Table VII. Specifically, we go long

a version of the trading strategy restricted to the top quartile of firms based on leverage

and go short the corresponding low leverage (bottom quartile) version. The resulting in-

sample alpha is 51 basis points per month with a t-statistic of 2.93. The out-of-sample

estimate still generates a statistically-significant 36 basis points per month.

V.A.1. Conditional Attribution

Give that beta is moving with 𝐶𝑜𝐵𝐴𝑅, we also estimate conditional performance

attribution regressions (that is, we allow for the possibility that portfolio betas and

expected market and factor returns comove in the time series). The last rows of Table III

Panel A and Panel B report the results of those regressions. We find that the long-run

reversal of beta-arbitrage profits remains; there is again an economically large reversal of

beta-arbitrage profits when 𝐶𝑜𝐵𝐴𝑅 is high. Figure 5 plots the conditional security market

line in the short and long-run as a function of lagged 𝐶𝑜𝐵𝐴𝑅. It is easy to see from the

figure our result that beta expansion and destabilization go hand-in-hand: the range of

average beta across the 20 beta-portfolios is much larger during high 𝐶𝑜𝐵𝐴𝑅 periods than

in low 𝐶𝑜𝐵𝐴𝑅 periods.

V.B. Low Limits to Arbitrage

26

We interpret our findings as consistent with arbitrage activity facilitating the correction

of the slope of the security market line in the short run. However, in periods of crowded

trading, arbitrageurs can cause price overshooting. In Table X, we exploit cross-sectional

heterogeneity to provide additional support for our interpretation. All else equal,

arbitrageurs prefer to trade stocks with low idiosyncratic volatility (to reduce tracking

error), high liquidity (to facilitate opening/closing of the position), and large capitalization

(to increase strategy capacity). As a consequence, we split our sample along each of these

dimensions. In particular, we rank stocks into quartiles based on the variable in question

(as of the beginning of the holding period); we label the quartile with the weakest limits

to arbitrage as “Low LTA” and the quartile with the strongest limits to arbitrage as

“High LTA.” Our focus is on the long-run reversal associated with periods of high 𝐶𝑜𝐵𝐴𝑅.

The first two columns report results based on market capitalization, the third and

fourth based on idiosyncratic volatility, and the final two based on liquidity. The first

column of each pair shows the difference in four-factor alpha to the beta arbitrage strategy

between high 𝐶𝑜𝐵𝐴𝑅 periods and low 𝐶𝑜𝐵𝐴𝑅 periods in months 1-6 while the second

column shows the difference occurring in year 3.

𝐶𝑜𝐵𝐴𝑅 is an economically stronger predictor of abnormal returns to beta-arbitrage

strategies in months 1-6 for large stocks, stocks with low idiosyncratic volatility, and those

with high liquidity. Two of the estimates are statistically significant, and we can easily

reject the null hypothesis that the three estimates are jointly zero (the average value is

0.87 with a t-statistic of 2.73). For each of the three proxies for low limits to arbitrage,

we also find economically and statistically significant differences in the predictability of

year 3 returns. In summary, Table X confirms that our effect is stronger among stocks

with weaker limits of arbitrage, exactly where one expects arbitrageurs to play a more

important role.

V.C. Time-series and Cross-sectional Variation in Fund Exposures

27

We next use our novel measure of beta-arbitrage activity to understand time-series and

cross-sectional variation in the performance of long-short/market-neutral hedge funds,

typically considered to be the classic example of smart money; as well as active mutual

funds, who are subject to more stringent leverage and short-sale constraints. Table XI

reports estimates of panel regressions of monthly fund returns on the Fama-French-

Carhart four-factor model augmented with the beta-arbitrage factor of Frazzini and

Pedersen (2014). In particular, we allow the coefficient on the Frazzini-Pedersen betting-

against-beta (BAB) factor to vary as a function of 𝐶𝑜𝐵𝐴𝑅, a fund’s AUM, and the

interaction between these two variables. To capture variation in a fund’s AUM, we create

a dummy-variable, 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘, that takes the value of zero if the fund is in the smallest-

AUM tercile (within the active mutual fund or long-short equity hedge fund industry,

depending on the returns being analyzed) in the previous month, one if it is in the middle

tercile, and two otherwise. The first two columns analyze hedge fund returns while the

last two columns analyze active mutual fund returns.

We find that the typical long-short equity hedge fund increases their exposure to

the BAB factor when 𝐶𝑜𝐵𝐴𝑅 is relatively high. For the 20% of the sample period that is

associated with the lowest values of 𝐶𝑜𝐵𝐴𝑅, the typical hedge fund’s BAB loading is

-0.08. This loading increases by 0.022 for each increment in 𝐶𝑜𝐵𝐴𝑅 rank. (It is noteworthy

that the average long-short hedge fund is loading negatively on the BAB factor – i.e., on

average, funds are tilting towards high beta stocks.)

Adding the interaction with AUM reveals that the ability of hedge funds to time

beta-arbitrage strategies is decreasing in the size of the fund’s assets under management.

These findings seem reasonable as we would expect large funds to be unable to time a

beta-arbitrage strategy as easily as smaller (and presumably nimbler) funds.

The typical small fund’s exposure increases by 0.035 for each increase in 𝐶𝑜𝐵𝐴𝑅

rank. Thus, when 𝐶𝑜𝐵𝐴𝑅 is in the top quintile, the typical small hedge fund’s BAB loading

is 0.048. In contrast, large hedge funds’ BAB loading moves from -0.118 to only 0.022

28

across the five 𝐶𝑜𝐵𝐴𝑅 quintiles, a much smaller increase in exposure to beta arbitrage.

Indeed, when 𝐶𝑜𝐵𝐴𝑅 is high, small hedge funds have loadings on BAB that are more than

twice as large.

As can be seen from columns 3 and 4, there is a vastly different pattern in the

market exposures of mutual funds. To start, mutual funds have an average market beta

that is larger than one, and hence unsurprisingly load negatively on the BAB factor. This

suggests that mutual funds on average overweight high beta stocks – consistent with the

intuition that most mutual funds are leverage-constrained. Second, none of the

interactions are statistically significant. In particular, mutual funds’ loadings on market

risk do not vary with 𝐶𝑜𝐵𝐴𝑅, our proxy for the strategy’s crowdedness.

V.D. Fresh versus Stale Beta

Though beta-arbitrage activity may cause the beta spread to vary through time, for a

feedback loop to occur, beta arbitrageurs must base their strategies on fresh estimates of

beta rather than on stale estimates. (Note that the autocorrelation in a stock’s market

beta is far less than one.) Consistent with this claim, we show that our predictability

results decay as a function of beta staleness.

We repeat the previous analysis of section IV.B, but replacing our fresh beta

estimates (measured over the most recent year) with progressively staler ones. In

particular, we estimate betas in each of the five years prior to the formation year. As a

consequence, both the resulting beta strategy and the associated 𝐶𝑜𝐵𝐴𝑅 are different for

each degree of beta staleness. For each of these six beta strategies, we regress the four-

factor alpha of the strategy in months one-six and year three on its corresponding 𝐶𝑜𝐵𝐴𝑅.

Figure 6 plots the resulting regression coefficients (results for months 1-6 plotted

with a blue square and results for year 3 plotted with a red circle) as a function of the

degree of staleness of beta. The baseline results with the most recent beta are simply the

corresponding figures from Table V. We find that both the short-run and long-run

29

predictability documented in section IV.B decay as the beta signal becomes more and

more stale. Indeed, strategies using beta estimates that are five years old display no

predictability. These results are consistent with the feedback loop we propose.24

VI. Conclusion

We study the response of arbitrageurs to a flat security market line. Using an approach

to measuring arbitrage activity first introduced by Lou and Polk (2014), we document

booms and busts in beta arbitrage. Specifically, we find that when arbitrage activity is

relatively low, abnormal returns on beta-arbitrage strategies take much longer to

materialize, appearing only two to three years after putting on the trade. In sharp

contrast, when arbitrage activity is relatively high, abnormal returns on beta-arbitrage

strategies occur relatively quickly, within the first six months of the trade. These strong

abnormal returns then revert over the next three years. Thus, our findings are consistent

with arbitrageurs exacerbating this time-variation in the expected return to beta

arbitrage.

We provide evidence on a novel positive feedback channel for beta-arbitrage

activity. Since the typical firm is levered and given the mechanical effects of leverage on

equity beta (Modigliani and Miller 1958), buying low-beta stocks and selling high-beta

stocks may cause the cross-sectional spread in betas to increase. We show that this beta

expansion occurs when beta-arbitrage activity is high and particularly so when stocks

typically traded by beta arbitrageurs are highly levered. Thus, beta arbitrageurs may

actually increase their bets when the profitability of the strategy has decreased. Indeed,

we find that the short-run abnormal returns to high-leverage beta-arbitrage stocks more

than triples before reverting in the long run.

24 Figure A1 documents that the unconditional CAPM alpha for a beta-arbitrage strategy also significantly

declines with beta staleness.

30

Interestingly, the unconditional four-factor alpha of a value-weight beta-arbitrage

strategy over our 1970-2010 sample is close to zero, much lower than the positive value

one finds for earlier samples. Thus, it seems that arbitrageurs’ response to Black, Jensen,

and Scholes’s (1972) famous finding has been right on average. However, our conditional

analysis reveals rich time-series variation that is consistent with the general message of

Stein (2009): Arbitrage activity faces a significant coordination problem for unanchored

strategies that have positive feedback characteristics.

31

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Table I: Summary Statistics

This table provides characteristics of “𝐶𝑜𝐵𝐴𝑅,” the excess comovement among low beta stocks over the period

1970-2013 (we then examine beta arbitrage returns in the following three years). At the end of each month,

all stocks are sorted into deciles based on their market beta calculated using daily returns in the past 12

months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. Pairwise

partial return correlations (controlling for the Fama-French three factors) for all stocks in the bottom beta

decile are computed based on weekly stock returns in the previous 12 months. 𝐶𝑜𝐵𝐴𝑅 is the average pair-

wise correlation between any two stocks in the low-beta decile in year 𝑡. 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 is the smoothed inflation

rate used by Cohen, Polk, and Vuolteenaho (2005) who apply an exponentially weighted moving average

(with a half-life of 36 months) to past log growth rates of the producer price index. 𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 is the sentiment

index proposed by Wurgler and Baker (2006, 2007). 𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡 is the beta-weighted standard deviation

of analysts’ long-term growth rate forecasts, as used in Hong and Sraer (2016). 𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 is the difference

between the LIBOR rate and the US Treasury bill rate. Panel A reports the summary statistics of these

variables. Panel B shows the time-series correlations among these variables for the entire sample period.

Panel A: Summary Statistics

Variable N Mean Std. Dev. Min Max

𝐶𝑜𝐵𝐴𝑅 528 0.105 0.026 0.037 0.203

Inflation 528 0.003 0.002 0.000 0.008

Sentiment 528 0.012 0.941 -2.325 3.076

Disagreement 384 -0.013 1.013 -1.277 3.593

Ted Spread 336 0.621 0.438 0.118 3.353

Panel B: Correlation

𝐶𝑜𝐵𝐴𝑅 Inflation Sentiment Disagreement Ted Spread

𝐶𝑜𝐵𝐴𝑅 1

Inflation -0.324 1

Sentiment 0.013 -0.394 1

Disagreement 0.336 -0.162 0.187 1

Ted Spread 0.267 0.241 -0.034 -0.181 1

Table II: Determinants of 𝐶𝑜𝐵𝐴𝑅

This table reports regressions of 𝐶𝑜𝐵𝐴𝑅, described in Table I, on lagged variables plausibly linked to arbitrage

activity in the post-1994 period (constrained by the availability of the hedge fund AUM data). At the end of

each month, all stocks are sorted into deciles based on their market beta calculated using daily returns in the

past 12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of

the regression equation five lags of the excess market return, in addition to the contemporaneous excess

market return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. The

dependent variable in the regressions, 𝐶𝑜𝐵𝐴𝑅, is the average pairwise partial weekly return correlation in the

low-beta decile over 12 months. 𝐼𝑛𝑠𝑡 𝑂𝑤𝑛 is the aggregate institutional ownership of the low-beta decile,

𝐴𝑈𝑀 is the logarithm of the total assets under management of long-short equity hedge funds (detrended).

𝐵𝐴𝐵 𝐴𝑙𝑝ℎ𝑎 is the realized four-factor alpha of Frazzini and Pedersen’s BAB factor. 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 is the smoothed

inflation rate used by Cohen, Polk, and Vuolteenaho (2005) who apply an exponentially weighted moving

average (with a half-life of 36 months) to past log growth rates of the producer price index. 𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 is the

sentiment index proposed by Wurgler and Baker (2006, 2007). 𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡 is the beta-weighted standard

deviation of analysts’ long-term growth rate forecasts, as used in Hong and Sraer (2012). 𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 is the

difference between the LIBOR rate and the US Treasury bill rate. We also include in the regression the

Pastor-Stambaugh liquidity factor (PS Liquidity). A trend dummy is included in all regression specifications.

All independent variables are divided by their corresponding standard deviation, so that the coefficient

represents the effect of a one-standard-deviation change in the independent variable on 𝐶𝑜𝐵𝐴𝑅. Standard

errors are shown in brackets. *, **, *** denote significance at the 10%, 5%, and 1% level, respectively.

DepVar 𝐶𝑜𝐵𝐴𝑅𝑡

[1] [2]

𝐼𝑛𝑠𝑡 𝑂𝑤𝑛𝑡−1 0.026*** 0.006**

[0.003] [0.003]

𝐴𝑈𝑀𝑡−1 0.004** 0.003**

[0.002] [0.002]

𝐵𝐴𝐵 𝐴𝑙𝑝ℎ𝑎𝑡−1 0.000 0.004***

[0.001] [0.001]

𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛𝑡 -0.005

[0.003]

𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡𝑡 0.003

[0.002]

𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡𝑡 0.011***

[0.002]

𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑𝑡 0.016***

[0.002]

𝑃𝑆 𝐿𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦𝑡 0.006*** 0.010***

[0.002] [0.001]

TREND Yes Yes

Adj-R2 0.350 0.529

No. Obs. 228 228

Table III: Forecasting Beta-arbitrage Returns with 𝐶𝑜𝐵𝐴𝑅

This table reports returns to the beta arbitrage strategy as a function of lagged 𝐶𝑜𝐵𝐴𝑅. At the end of each

month, all stocks are sorted into deciles based on their market beta calculated using daily returns in the past

12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. All months

are then classified into five groups based on 𝐶𝑜𝐵𝐴𝑅, the average pairwise partial weekly return correlation in

the low-beta decile over the past 12 months. Reported below are the returns to the beta arbitrage strategy

(i.e., to go long the value-weight low-beta decile and short the value-weighted high-beta decile) in each of the

three years after portfolio formation during 1970 to 2013, following low to high 𝐶𝑜𝐵𝐴𝑅. Panels A and B

report, respectively, the average monthly three-factor alpha and Carhart four-factor alpha of the beta

arbitrage strategy. “5-1” is the difference in monthly returns to the long-short strategy following high vs. low

𝐶𝑜𝐵𝐴𝑅; “5-1 Conditional” is the difference in conditional abnormal returns (i.e., allowing for risk loadings to

vary as a function of 𝐶𝑜𝐵𝐴𝑅) following high vs. low 𝐶𝑜𝐵𝐴𝑅. T-statistics, shown in parentheses, are computed

based on standard errors corrected for serial-dependence up to 24 lags. 5% statistical significance is indicated

in bold.

Panel A: Fama-French-Adjusted Beta-arbitrage Returns

Months 1-6 Months 7-12 Year 2 Year 3 Years 2&3

Rank Obs. Est. t-stat Est. t-stat Est. t-stat Est. t-stat Est. t-stat

1 105 0.58% (2.20) 0.58% (2.41) 0.65% (2.40) 0.88% (3.41) 0.73% (2.72)

2 106 -0.16% (-0.53) 0.56% (1.91) 0.74% (2.70) 0.17% (0.82) 0.49% (2.36)

3 106 -0.35% (-1.19) 0.20% (0.76) 0.41% (1.57) 0.55% (2.29) 0.49% (2.24)

4 106 -0.19% (-0.69) 0.33% (1.18) -0.09% (-0.28) -0.01% (-0.02) -0.03% (-0.11)

5 105 1.47% (3.62) 0.37% (0.81) 0.02% (0.04) -0.60% (-1.68) -0.28% (-1.59)

5-1 0.89% (1.89) -0.21% (-0.41) -0.63% (-1.10) -1.48% (-3.17) -1.01% (-2.66)

5-1(Cond) 0.30% (0.74) -0.62% (-1.32) -0.85% (-1.54) -1.34% (-3.06) -1.05% (-3.08)

Panel B: Four-Factor Adjusted Beta-arbitrage Returns

Months 1-6 Months 7-12 Year 2 Year 3 Years 2&3

Rank Obs. Est. t-stat Est. t-stat Est. t-stat Est. t-stat Est. t-stat

1 105 0.33% (1.21) 0.29% (1.17) 0.40% (1.49) 0.59% (2.22) 0.45% (1.67)

2 106 -0.36% (-1.28) 0.19% (0.64) 0.46% (1.69) -0.03% (-0.13) 0.25% (1.06)

3 106 -0.57% (-1.97) -0.08% (-0.32) 0.08% (0.27) 0.30% (1.19) 0.21% (0.81)

4 106 -0.49% (-1.79) -0.01% (-0.03) -0.41% (-1.10) -0.35% (-0.85) -0.36% (-0.98)

5 105 1.08% (2.63) 0.14% (0.33) -0.14% (-0.31) -0.93% (-2.66) -0.53% (-2.58)

5-1 0.75% (1.57) -0.15% (-0.30) -0.54% (-1.02) -1.52% (-3.33) -0.99% (-2.52)

5-1 (Cond) 0.08% (0.20) -0.74% (-1.63) -1.02% (-2.15) -1.68% (-4.08) -1.33% (-3.89)

Table IV: Robustness Checks

This table reports returns to the beta arbitrage strategy as a function of lagged 𝐶𝑜𝐵𝐴𝑅. At the end of each

month, all stocks are sorted into deciles based on their market beta calculated using daily returns in the past

12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. All months

are then classified into five groups based on 𝐶𝑜𝐵𝐴𝑅, the average pairwise partial weekly return correlation in

the low-beta decile over the past 12 months. Reported below is the difference in four-factor alpha to the beta

arbitrage strategy between high 𝐶𝑜𝐵𝐴𝑅 periods and low 𝐶𝑜𝐵𝐴𝑅 periods. Year zero is the beta portfolio

ranking period. Row 1 shows the baseline results which are also reported in Table III. Row 2 shows the same

analysis for the earlier sample (1927-1969) as a placebo test. In rows 3 and 4, we exclude the tech bubble

crash and the recent financial crisis from our sample. In rows 5-8, we rank all months based on smoothed

past inflation (Cohen, Polk, and Vuolteenaho, 2005), sentiment index (Wurgler and Baker, 2006), aggregate

analyst forecast dispersion (Hong and Sraer, 2014), and Ted Spread (Frazzini and Pedersen, 2014),

respectively. In row 9, we also control for the UMD factor in computing 𝐶𝑜𝐵𝐴𝑅. In row 10, we control for

both large- and small-cap HML in computing 𝐶𝑜𝐵𝐴𝑅. In row 11, we control for the Fama-French five factor

model that adds profitability and investment to their three-factor model. In row 12, we perform the entire

analysis on an industry-adjusted basis by sorting stocks into beta deciles within industries. In row 13, we

instead measure the correlation between the high and low-beta portfolios, with a low correlation indicating

high arbitrage activity. In Rows 14 and 15, we examine upside and downside 𝐶𝑜𝐵𝐴𝑅, as distinguished by the

median low-beta portfolio return. In Rows 16-22, we replace 𝐶𝑜𝐵𝐴𝑅 with residual 𝐶𝑜𝐵𝐴𝑅 from a time-series

regression where we purge from 𝐶𝑜𝐵𝐴𝑅 variation linked to, respectively, 𝐶𝑜𝑀𝑂𝑀 and 𝐶𝑜𝑉𝑎𝑙𝑢𝑒 (Lou and

Polk, 2017), the average pair-wise correlation in the market, the lagged 36-month volatility of the BAB factor

(Frazzini and Pedersen, 2013), market volatility over the past 24 months, a trend, and lagged 𝐶𝑜𝐵𝐴𝑅 (where

we hold the stocks in the low-beta decile constant but calculate 𝐶𝑜𝐵𝐴𝑅 using returns from the previous year).

In Row 23, abnormal returns to the beta arbitrage strategy is calculated relative to the Fama-French 5 factor

model plus the momentum factor. T-statistics, shown in parentheses, are computed based on standard errors

corrected for serial-dependence up to 24 lags. 5% statistical significance is indicated in bold.

Four-Factor Adjusted Beta-arbitrage Returns

Year 3 Years 2-3

Estimate t-stat Estimate t-stat

Subsamples

Full Sample: 1970-2010 -1.52% (-3.33) -0.99% (-2.52)

Early sample: 1927-1969 0.19% (0.41) 0.22% (0.68)

Excluding 2001 -1.56% (-3.44) -0.99% (-2.24)

Excluding 2007-2009 -1.48% (-3.06) -0.94% (-2.29)

Other predictors of beta-arbitrage returns

Inflation 0.13% (0.24) 0.27% (0.54)

Sentiment -0.11% (-0.17) 0.07% (0.11)

Disagreement 0.67% (1.03) 0.86% (1.83)

Ted Spread -0.21% (-0.41) -0.18% (-0.45)

Alternative definitions of 𝐶𝑜𝐵𝐴𝑅

Controlling for UMD -1.59% (-3.70) -1.09% (-2.65)

Controlling for Large/Small-Cap HML -1.53% (-3.35) -0.97% (-2.47)

Controlling for FF Five Factors -1.32% (-2.67) -1.00% (-2.54)

Controlling for Industry Factors -1.29% (-3.02) -0.93% (-2.37)

Correl btw High and Low Beta Stocks -0.88% (-2.33) -0.81% (-2.19)

Upside 𝐶𝑜𝐵𝐴𝑅 -0.99% (-2.29) -0.83% (-2.26)

Downside 𝐶𝑜𝐵𝐴𝑅 -0.30% (-0.67) 0.23% (0.94)

Residual 𝐶𝑜𝐵𝐴𝑅

Controlling for CoMomentum -1.65% (-3.78) -1.10% (-2.91)

Controlling for CoValue -1.57% (-3.63) -1.00% (-2.52)

Controlling for MKT CORR -1.82% (-4.21) -1.21% (-2.93)

Controlling for Vol(BAB) -1.66% (-3.81) -1.07% (-2.74)

Controlling for Mktvol24 -1.50% (-3.24) -0.98% (-2.57)

Controlling for Trend -1.51% (-3.31) -0.98% (-2.49)

Controlling for Pre-formation CoBAR -1.56% (-3.27) -1.13% (-2.85)

Beta arbitrage alpha based on

Controlling for FF5+MOM -1.46% (-3.52) -1.03% (-2.99)

Table V: Regression Analysis

This table reports returns to the beta arbitrage strategy as a function of lagged 𝐶𝑜𝐵𝐴𝑅. At the end of each

month, all stocks are sorted into deciles based on their market beta calculated using daily returns in the past

12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. The dependent

variable is the four-factor alpha of the beta arbitrage strategy (i.e., a portfolio that is long the value-weight

low-beta decile and short the value-weighted high-beta decile). The main independent variable is 𝐶𝑜𝐵𝐴𝑅, the

average pairwise partial weekly three-factor residual correlation within the low-beta decile over the past 12

months. We also include in the regression exponentially weighted moving average past inflation (Cohen, Polk,

and Vuolteenaho, 2005), a sentiment index (Wurgler and Baker, 2006), aggregate analyst forecast dispersion

(Hong and Sraer, 2016), Ted Spread—the difference between the LIBOR rate and the US Treasury bill rate,

the ValueSpread—the spread in log book-to-market-ratios across the low-beta and high-beta deciles, and

market volatility over the past 24 months. The first three columns examine returns to the beta arbitrage

strategy in months 1-6, and the next three columns examine the returns in year 3 after portfolio formation.

We report results based on Carhart four-factor adjustments. T-statistics, shown in brackets, are computed

based on standard errors corrected for serial-dependence up to 12 lags. *, **, *** denote significance at the

10%, 5%, and 1% level, respectively.

DepVar Four-Factor Alpha to the Beta Arbitrage Strategy

Months 1-6 Year 3

[1] [2] [3] [4] [5] [6]

𝐶𝑜𝐵𝐴𝑅 0.145** 0.177*** 0.152* -0.195*** -0.154*** -0.277*** [0.067] [0.072] [0.087] [0.054] [0.053] [0.079]

𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 1.384 3.316 1.685* 2.793 [1.195] [2.253] [0.962] [3.358]

𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 0.004** -0.003 0.002 0.002 [0.002] [0.003] [0.002] [0.004]

𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡 0.006*** 0.005

[0.003] [0.003]

𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 -0.008 0.001

[0.007] [0.008]

𝑉𝑎𝑙𝑢𝑒𝑆𝑝𝑟𝑒𝑎𝑑 0.002 0.004 0.001 -0.002

[0.005] [0.004] [0.003] [0.003]

𝑀𝑘𝑡𝑣𝑜𝑙24 0.065 -0.092 0.118 0.045

[0.106] [0.148] [0.150] [0.214]

Adj-R2 0.024 0.048 0.085 0.094 0.128 0.154

No. Obs. 528 528 336 528 528 336

Table VI: Predicting the Security Market Line

This table reports regressions of the intercept and slope of the security market line on lagged 𝐶𝑜𝐵𝐴𝑅. At the

end of each month, all stocks are sorted into vigintiles based on their market beta calculated using daily

returns in the past 12 months. To account for illiquidity and non-synchronous trading, we include on the

right-hand side of the regression equation five lags of the excess market return, in addition to the

contemporaneous excess market return. The pre-ranking beta is simply the sum of the six coefficients from

the OLS regression. We then estimate security market lines using monthly portfolio returns in months 1-6,

based on these 20 portfolios formed in each period. The post-ranking betas are calculated by regressing each

of the 20 portfolios’ value-weighted monthly returns on the corresponding market return. Following Fama

and French (1992), we use the entire sample to compute post-ranking betas. The dependent variable in Panel

A is the intercept of the SML, while that in Panel B is the slope of the SML. The main independent variable

is 𝐶𝑜𝐵𝐴𝑅, the average pairwise partial weekly three-factor residual correlation within the low-beta decile over

the past 12 months. We also include in the regressions smoothed inflation, sentiment index, aggregate analyst

forecast dispersion, and Ted Spread. Other (unreported) control variables include the contemporaneous

market excess return, SMB return, and HML return. Standard errors, shown in brackets, are computed based

on standard errors corrected for serial-dependence. *, **, *** denote significance at the 10%, 5%, and 1%

level, respectively.

Panel A: DepVar = Intercept of SML

Months 1-6 Year3

𝐶𝑜𝐵𝐴𝑅 0.171*** 0.226*** 0.134** -0.266*** -0.189*** -0.226*** [0.070] [0.051] [0.065] [0.077] [0.067] [0.087]

𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 2.585*** 5.728*** 0.422 1.069 [0.768] [1.670] [1.229] [2.157]

𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 0.006*** 0.004 -0.001 -0.004 [0.001] [0.003] [0.002] [0.004]

𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡 0.004*** 0.001 [0.002] [0.003]

𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 -0.011*** 0.004 [0.004] [0.004]

Adj-R2 0.044 0.366 0.511 0.119 0.371 0.471

No. Obs. 528 528 336 528 528 336

Panel B: DepVar = Slope of SML

Months 1-6 Year3

𝐶𝑜𝐵𝐴𝑅 -0.337*** -0.215*** -0.130** 0.289*** 0.207*** 0.203** [0.088] [0.050] [0.066] [0.096] [0.069] [0.094]

𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 -2.465*** -5.716*** -0.833 -0.006 [0.792] [1.706] [1.191] [2.583]

𝑆𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡 -0.006*** -0.005* 0.001 0.005 [0.001] [0.003] [0.002] [0.004]

𝐷𝑖𝑠𝑎𝑔𝑟𝑒𝑒𝑚𝑒𝑛𝑡 -0.003* 0.000 [0.002] [0.004]

𝑇𝑒𝑑 𝑆𝑝𝑟𝑒𝑎𝑑 0.012*** -0.003 [0.004] [0.004]

Adj-R2 0.092 0.668 0.732 0.104 0.468 0.498

No. Obs. 528 528 336 528 528 336

Table VII: Smarter Beta-Arbitrage Strategies

This Table reports monthly returns to a “smarter” beta-arbitrage strategy that exploits the time-varying

overreaction and subsequent reversal in standard beta arbitrage strategies. Specifically, we first time the

standard beta-arbitrage strategy using current 𝑪𝒐𝑩𝑨𝑹 . If 𝑪𝒐𝑩𝑨𝑹 is above the 80th percentile, we go long

the long-short beta-arbitrage strategy studied in Table III for the next six months. Otherwise, we short that

portfolio over that time period. In addition, if 𝑪𝒐𝑩𝑨𝑹 from three years ago is below the 20th percentile, we

are long for the next twelve months the long-short beta-arbitrage strategy based on beta estimates from three

years ago. Otherwise, we short that portfolio, again for the next twelve months. In Panel A, the percentile

break points of the 𝑪𝒐𝑩𝑨𝑹 distribution are identified using the entire 𝑪𝒐𝑩𝑨𝑹 time series (thus an in-sample

analysis). In Panel B, the percentile break points are identify based solely on its prior distribution (an out-

of-sample analysis); we skip the first three years of our sample to compute the initial distribution. In the full

specification of risk adjustment, we use a seven-factor model that includes: the Fama and French (2015) five-

factors (market, size, value, investment and profitability), the Carhart momentum factor, and Frazzini and

Pedersen’s (2014) betting-against beta(BAB) factor. 5% statistical significance is indicated in bold.

ALPHA RM-RF SMB HML UMD RMW CMA BAB

Panel A: In-Sample

0.53% 0.52 0.10 0.04 -0.15

(2.65) (10.62) (1.52) (0.37) (-2.07)

0.49% 0.51 0.19 0.06 -0.15 0.16 -0.12

(2.34) (10.38) (2.43) (0.55) (-2.10) (1.20) (-0.58)

0.62% 0.56 0.26 0.20 -0.08 0.39 0.02 -0.45

(2.98) (11.69) (3.38) (1.62) (-1.10) (2.96) (0.10) (-5.16)

Panel B: Out-of-Sample

0.51% 0.52 0.10 -0.02 -0.20

(2.62) (11.16) (1.51) (-0.25) (-2.86)

0.51% 0.51 0.17 0.04 -0.19 0.10 -0.19

(2.46) (10.64) (2.14) (0.36) (-2.76) (0.66) (-0.95)

0.63% 0.55 0.24 0.18 -0.12 0.32 -0.06 -0.44

(3.10) (11.82) (3.08) (1.48) (-1.71) (2.29) (-0.29) (-5.30)

Table VIII: Beta Expansion, Time-Series Analysis

This table examines time-series beta expansion associated with arbitrage trading. Panel A reports the baseline

regression. The dependent variable is the beta spread between the high-beta and low-beta deciles (ranked in

year 𝑡) in year 𝑡+1. 𝐶𝑜𝐵𝐴𝑅 is the average pairwise partial weekly three-factor residual correlation within the

low-beta decile over the past 12 months. 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 is a quintile dummy based on the average value-weighted

book leverage of the bottom and top beta deciles. We also include in the regression an interaction term

between 𝐶𝑜𝐵𝐴𝑅 and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. Panel B reports a battery of robustness checks. The dependent variable in

all rows is the beta spread between the high-beta and low-beta deciles in year 𝑡+1. Reported below is the

coefficient on the interaction of 𝐶𝑜𝐵𝐴𝑅 and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. Row 1 shows the baseline results which are also

reported in Table III. In Rows 2 and 3, we exclude the tech bubble crash and the recent financial crisis from

our sample. In Row 4, we also control for the UMD factor in computing 𝐶𝑜𝐵𝐴𝑅. In Row 5, we control for

both large- and small-cap HML in computing 𝐶𝑜𝐵𝐴𝑅. In Row 6, we control for the Fama-French five factor

model that adds profitability and investment to their three-factor model. In Row 7, we perform the entire

analysis on an industry-adjusted basis by sorting stocks into beta deciles within industries. In Row 8, we

instead measure the correlation between the high and low-beta portfolios, with a low correlation indicating

high arbitrage activity. In Rows 9 and 10, we examine upside and downside 𝐶𝑜𝐵𝐴𝑅, as distinguished by the

median low-beta portfolio return. In Rows 11-16, we replace 𝐶𝑜𝐵𝐴𝑅 with residual 𝐶𝑜𝐵𝐴𝑅 from a time-series

regression where we purge from 𝐶𝑜𝐵𝐴𝑅 variation linked to, respectively, 𝐶𝑜𝑀𝑂𝑀 and 𝐶𝑜𝑉𝑎𝑙𝑢𝑒 (Lou and

Polk, 2014), the average pair-wise correlation in the market, the lagged 36-month volatility of the BAB factor

(Frazzini and Pedersen, 2014), market volatility over the past 24 months, a trend, and lagged 𝐶𝑜𝐵𝐴𝑅 (where

we hold the stocks in the low-beta decile constant but calculate 𝐶𝑜𝐵𝐴𝑅 using returns from the previous year).

Standard errors are shown in brackets. *, **, *** denote significance at the 10%, 5%, and 1% level,

respectively.

Panel A: Baseline Regression

DepVar 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑𝑡+1

[1] [2]

𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑 0.234*** 0.232***

[0.060] [0.058]

𝐶𝑜𝐵𝐴𝑅 1.148** 0.163

[0.527] [0.636]

𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 -0.035***

[0.015]

𝐶𝑜𝐵𝐴𝑅 ∗ 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 0.408***

[0.117]

Adj-R2 0.083 0.103

No. Obs. 528 528

Panel B: Robustness Checks

DepVar = 𝐵𝑒𝑡𝑎𝑆𝑝𝑟𝑒𝑎𝑑𝑡+1

Estimate Std. Dev.

Subsamples

Full Sample:1970-2016 0.408*** [0.117]

Excluding 2001 0.413*** [0.113]

Excluding 2007-2009 0.311** [0.140]

Alternative definitions of COBAR

Controlling for UMD 0.433*** [0.117]

Controlling for Large/Small-Cap HML 0.357*** [0.127]

Controlling for FF Five Factors 0.429*** [0.132]

Controlling for Industry Factors 0.473*** [0.128]

Correl btw High and Low Beta Stocks 0.142*** [0.040]

Upside 𝐶𝑜𝐵𝐴𝑅 0.513*** [0.172]

Downside 𝐶𝑜𝐵𝐴𝑅 0.281** [0.143]

Residual 𝐶𝑜𝐵𝐴𝑅

Controlling for 𝐶𝑜𝑀𝑂𝑀 0.235** [0.115]

Controlling for 𝐶𝑜𝑉𝑎𝑙𝑢𝑒 0.240** [0.118]

Controlling for MKT CORR 0.382*** [0.122]

Controlling for Vol(BAB) 0.382*** [0.110]

Controlling for Mktvol24 0.261** [0.118]

Controlling for Trend 0.279** [0.119]

Controlling for Pre-formation 𝐶𝑜𝐵𝐴𝑅 0.433*** [0.120]

Table IX: Beta Expansion, Cross-Sectional Analysis

This table reports panel regressions of post-ranking stock beta on lagged 𝐶𝑜𝐵𝐴𝑅. At the end of each month,

all stocks are sorted into deciles based on their market beta calculated using daily returns in the past 12

months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. The dependent

variable is the post-ranking stock beta from year 𝑡 to 𝑡+1 (non-overlapping periods). The main independent

variable is lagged 𝐶𝑜𝐵𝐴𝑅, the average pairwise excess weekly return correlation in the low-beta decile over

the past 12 months. 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 is the difference between a stock’s pre-ranking beta and the average pre-ranking

beta in year 𝑡. 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 is the book leverage of the firm, measured in year 𝑡. We also include all double and

triple interaction terms of 𝐶𝑜𝐵𝐴𝑅, 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, and 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒. Other (unreported) control variables include the

lagged firm size, book-to-market ratio, past one-year return, idiosyncratic volatility (over the prior year), and

past one-month return. Time-fixed effects are included in Columns 3 and 4. (Since 𝐶𝑜𝐵𝐴𝑅 is a time-series

variable, it is subsumed by the time dummies.) Standard errors, shown in brackets, are double clustered at

both the firm and year-month levels. *, **, *** denote significance at the 10%, 5%, and 1% level, respectively.

DepVar 𝑃𝑜𝑠𝑡 𝑅𝑎𝑛𝑘𝑖𝑛𝑔 𝐵𝑒𝑡𝑎𝑡+1 [1] [2] [3] [4]

𝐶𝑜𝐵𝐴𝑅 -0.873*** -0.823***

[0.154] [0.144]

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 0.281*** 0.262*** 0.264*** 0.236*** [0.029] [0.032] [0.027] [0.030]

𝐶𝑜𝐵𝐴𝑅 ∗ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 0.768*** 0.571* 0.533** 0.531** [0.268] [0.298] [0.243] [0.268]

𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 -0.002 -0.003* [0.002] [0.002]

𝐶𝑜𝐵𝐴𝑅 ∗ 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 0.008 0.019 [0.016] [0.015]

𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 ∗ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 -0.004 0.005 [0.005] [0.005]

𝐶𝑜𝐵𝐴𝑅 ∗ 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 ∗ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 0.210*** 0.087** [0.050] [0.044]

Time Fixed Effects No No Yes Yes

Adj-R2 0.266 0.270 0.323 0.325

No. Obs. 1,194,641 1,194,641 1,194,641 1,194,641

Table X: Limits to Arbitrage

This table reports returns to the beta arbitrage strategy as a function of lagged 𝐶𝑜𝐵𝐴𝑅 in various subsamples

ranked by proxies for limits to arbitrage (LTA). At the end of each month, all stocks are sorted into deciles

based on their market beta calculated using daily returns in the past 12 months. To account for illiquidity

and non-synchronous trading, we include on the right-hand side of the regression equation five lags of the

excess market return, in addition to the contemporaneous excess market return. The pre-ranking beta is

simply the sum of the six coefficients from the OLS regression. All months are then classified into five groups

based on 𝐶𝑜𝐵𝐴𝑅, the average pairwise partial return correlation in the low-beta decile over the past 12

months. Reported below is the difference in four-factor alpha to the beta arbitrage strategy between high

𝐶𝑜𝐵𝐴𝑅 periods and low 𝐶𝑜𝐵𝐴𝑅 periods. Year zero is the beta portfolio ranking period. “Low LTA”

corresponds to the subsample of stocks with low limits to arbitrage, and “high LTA” corresponds to the

subsample with high limits to arbitrage. “Low-High” is the difference in monthly portfolio alpha between the

two subsamples. We measure limits to arbitrage using three common proxies. In columns 1-2, we rank stocks

into quartiles based on market capitalization (as of the beginning of the holding period); we label the top

quartile as “Low LTA” and the bottom quartile as “High LTA.” In columns 3-4, we rank stocks into quartiles

based on idiosyncratic volatility with regard to the Carhart four-factor model (as of the beginning of the

holding period); we label the bottom quartile as “Low LTA” and the top quartile as “High LTA.” In columns

5-6, we rank stocks into quartiles based on the illiquidity measure of Amihud (2002) (as of the beginning of

the holding period); we label the bottom quartile as “Low LTA” and the top quartile as “High LTA.” T-

statistics, shown in parentheses, are computed based on standard errors corrected for serial-dependence up

to 24 lags. 5% statistical significance is indicated in bold.

Market Cap Idiosyncratic Volatility Illiquidity

Months 1-6 Year 3 Months 1-6 Year 3 Months 1-6 Year 3

Low LTA 0.82% -1.55% 0.93% -1.61% 0.82% -1.48%

(2.42) (-3.19) (2.74) (-3.84) (2.39) (-2.92)

High LTA -0.10% 0.10% 0.30% -0.14% -0.23% -0.11%

(-0.30) (0.18) (0.66) (-0.27) (-0.66) (-0.25)

Low-High 0.92% -1.65% 0.63% -1.47% 1.06% -1.38%

(2.41) (-2.82) (1.38) (-2.97) (2.73) (-3.03)

Table XI: Beta Arbitrage Timing Ability

This table reports regressions of monthly mutual funds and hedge funds returns on a conditional six-factor

model. At the end of each month, we track mutual funds and hedge funds’ performance in the next six

months. The dependent variable in columns (1) and (2) is the average monthly excess return of long-short

equity hedge funds over that six-month window. Columns (3) and (4) are similar but use the monthly

excess returns on actively-managed mutual funds instead. We attribute those returns to 𝑚𝑘𝑡𝑟𝑓, 𝑠𝑚𝑏, ℎ𝑚𝑙, 𝑢𝑚𝑑 and 𝐵𝐴𝐵, the corresponding returns on the Fama-French three factors augmented by a momentum

factor and the four-factor alpha of the portfolio return buying stocks from the lowest beta decile and

shorting stocks from the highest beta decile. We allow the loading on BAB to be a function of lagged

𝐶𝑜𝐵𝐴𝑅 (quintile ranks from 1 to 5) and 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘, the lagged cross-sectional ranking of the fund’s assets

under management. To compute 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘, we first rank all funds (within the respective hedge fund or

mutual fund subset) into three groups as of the previous month and then assign the value 2 if the fund is

in the highest group, 1 if the fund is in the middle group, and 0 if the fund is in the lowest group. Standard

errors, shown in bracket, are clustered at both the month and fund level. *, **, *** denote significance at

the 10%, 5%, and 1% level, respectively.

Equity Hedge Funds Equity Mutual Funds

[1] [2] [3] [4]

𝑚𝑘𝑡𝑟𝑓 0.354*** 0.354*** 1.029*** 1.029*** [0.055] [0.055] [0.011] [0.011]

𝑠𝑚𝑏 0.241*** 0.241*** 0.216*** 0.216*** [0.039] [0.039] [0.021] [0.021]

ℎ𝑚𝑙 -0.049 -0.049 0.079*** 0.079*** [0.036] [0.036] [0.021] [0.021]

𝑢𝑚𝑑 0.022 0.022 0.016* 0.016* [0.019] [0.019] [0.009] [0.009]

𝐵𝐴𝐵 -0.102*** -0.127*** -0.026* -0.016 [0.022] [0.028] [0.014] [0.019]

𝐶𝑜𝐵𝐴𝑅 0.001* 0.001** -0.000 -0.000 [0.000] [0.000] [0.000] [0.000]

𝐵𝐴𝐵 ∗ 𝐶𝑜𝐵𝐴𝑅 0.022*** 0.035*** 0.001 0.007 [0.008] [0.009] [0.005] [0.008]

𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘 -0.001** 0.000 [0.000] [0.000]

𝐵𝐴𝐵 ∗ 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘 0.0248* -0.010 [0.014] [0.012]

𝐶𝑜𝐵𝐴𝑅 ∗ 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘 -0.000 0.000 [0.000] [0.000]

𝐵𝐴𝐵 ∗ 𝐶𝑜𝐵𝐴𝑅 ∗ 𝑆𝑖𝑧𝑒𝑅𝑎𝑛𝑘 -0.013*** -0.006 [0.004] [0.004]

Adj-R2 0.018 0.018 0.690 0.690

No. Obs. 191,459 191,459 386,479 386,479

Figure 1: This figure shows the time series of the December observations of the 𝐶𝑜𝐵𝐴𝑅 measure. At the end

of each month, all stocks are sorted into deciles based on their market beta calculated using daily returns in

the past 12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand

side of the regression equation five lags of the excess market return, in addition to the contemporaneous

excess market return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression.

𝐶𝑜𝐵𝐴𝑅 is the average pairwise partial return correlation in the low-beta decile measured in the ranking period.

We begin our analysis in 1970, as that year was when the low-beta anomaly was first recognized by academics.

The time series average of 𝐶𝑜𝐵𝐴𝑅 is 0.11.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

𝐶𝑜𝐵𝐴𝑅 Time Series

Figure 2: This figure shows returns to the beta arbitrage strategy as a function of lagged 𝐶𝑜𝐵𝐴𝑅. At the end

of each month, all stocks are sorted into deciles based on their market beta calculated using daily returns in

the past 12 months. To account for illiquidity and non-synchronous trading, we include on the right-hand

side of the regression equation five lags of the excess market return, in addition to the contemporaneous

excess market return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression.

All months are then sorted into five groups based on 𝐶𝑜𝐵𝐴𝑅, the average pairwise weekly three-factor residual

correlation within the low-beta decile over the previous 12 months. The red curve shows the cumulative

Carhart four-factor alpha to the beta arbitrage strategy (i.e., a portfolio that is long the value-weight low-

beta decile and short the value-weighted high-beta decile) formed in high 𝐶𝑜𝐵𝐴𝑅 periods, the dotted blue

curve shows the cumulative Carhart four-factor alpha to the beta arbitrage strategy formed in periods of low

𝐶𝑜𝐵𝐴𝑅, and the grey curve shows the cumulative Carhart four-factor alpha to the beta arbitrage strategy

formed in periods of median 𝐶𝑜𝐵𝐴𝑅.

Cum

ula

tive

Four-

Fact

or

Alp

ha

Cum

ula

tive

Four-

Fact

or

Alp

ha

-5%

0%

5%

10%

15%

20%

25%

30%

-6 -4 -1 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59

Low CoBar Median CoBar High CoBar

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

0 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59

Low CoBar Median CoBar High CoBar

Figure 3: This figure shows the security market line as a function of lagged 𝐶𝑜𝐵𝐴𝑅. At the end of each month,

all stocks are sorted into vigintiles based on their market beta calculated using daily returns in the past 12

months. To account for illiquidity and non-synchronous trading, we include on the right-hand side of the

regression equation five lags of the excess market return, in addition to the contemporaneous excess market

return. The pre-ranking beta is simply the sum of the six coefficients from the OLS regression. We then

estimate two security market lines based on these 20 portfolios formed in each period: one SML using portfolio

returns in months 1-6, and the other using portfolio returns in year 3 after portfolio formation; the betas used

in these SML regressions are the corresponding post-ranking betas. The Y-axis reports the average monthly

excess returns to these 20 portfolios, and the X-axis reports the post-ranking betas of these portfolios. Beta

portfolios formed in high 𝐶𝑜𝐵𝐴𝑅 periods are depicted with a blue circle and fitted with a solid line, and those

formed in low 𝐶𝑜𝐵𝐴𝑅 periods are depicted with a red triangle and fitted with a dotted line. The top panel

shows average excess returns and betas to the beta-arbitrage strategy in months 1-6 after portfolio formation,

while the bottom panel shows average excess returns and betas in year 3 after portfolio formation.

-4%

-3%

-2%

-1%

0%

1%

2%

0 0.5 1 1.5 2 2.5

Low CoBar High CoBar

-1%

-1%

0%

1%

1%

2%

2%

0 0.5 1 1.5 2 2.5

Low CoBar High CoBar

post-ranking beta

post-ranking beta

Aver

age

Exce

ss R

eturn

in M

onth

s 1-6

A

ver

age

Exce

ss R

eturn

in Y

ear

3

Figure 4: This figure shows how the relation between 𝐶𝑜𝐵𝐴𝑅 and subsequent beta arbitrage returns varies

with firm leverage. At the end of each month, all stocks are sorted into deciles based on their market beta

calculated using daily returns in the past 12 months. To account for illiquidity and non-synchronous trading,

we include on the right-hand side of the regression equation five lags of the excess market return, in addition

to the contemporaneous excess market return. The pre-ranking beta is simply the sum of the six coefficients

from the OLS regression. All months are then sorted into five groups based on 𝐶𝑜𝐵𝐴𝑅, the average pairwise

weekly three-factor residual correlation within the low-beta decile over the previous 12 months. At the

beginning of the holding period, we sort stocks into four equal groups using book leverage. For each leverage

quartile, we compute the 𝐶𝑜𝐵𝐴𝑅 return spread – i.e., the difference in Carhart four-factor alpha to the beta

arbitrage strategy (i.e., a portfolio that is long the value-weight low-beta decile and short the value-weighted

high-beta decile) between high and low 𝐶𝑜𝐵𝐴𝑅 periods. The solid red curve shows the cumulative difference

in the 𝐶𝑜𝐵𝐴𝑅 return spread between the highest and lowest leverage quartiles over the five years after

portfolio formation. This difference in the 𝐶𝑜𝐵𝐴𝑅 return spread is 1.46%/month (t-statistic = 2.14) in year

one and is –0.52%/month (t-statistic = -2.41) in years four-five.

0%

5%

10%

15%

20%

25%

30%

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

High-Leverage versus Low-Leverage Firms

Figure 5: This figure shows the conditional security market line as a function of lagged 𝐶𝑜𝐵𝐴𝑅 (i.e., where

betas are allowed to vary with 𝐶𝑜𝐵𝐴𝑅). At the end of each month, all stocks are sorted into vigintiles based

on their market beta calculated using daily returns in the past 12 months. To account for illiquidity and non-

synchronous trading, we include on the right-hand side of the regression equation five lags of the excess

market return, in addition to the contemporaneous excess market return. The pre-ranking beta is simply the

sum of the six coefficients from the OLS regression. We then estimate two conditional security market lines

based on these 20 portfolios: one SML using portfolio returns in months 1-6, and the other using portfolio

returns in year 3 after portfolio formation; the betas used in these SML regressions are the corresponding

post-ranking betas. The Y-axis reports the average monthly excess returns to these 20 portfolios, and the X-

axis reports the post-ranking beta of these portfolios. Beta portfolios formed in high 𝐶𝑜𝐵𝐴𝑅 periods are

depicted with a blue circle and fitted with a solid line, and those formed in low 𝐶𝑜𝐵𝐴𝑅 periods are depicted

with a red triangle and fitted with a dotted line. The top panel shows average excess returns and betas to

the beta arbitrage strategy in months 1-6 after portfolio formation, while the bottom panel shows average

excess returns and betas in year 3 after portfolio formation.

-4%

-3%

-2%

-1%

0%

1%

2%

0 0.5 1 1.5 2 2.5

Low CoBar High CoBar

-1%

-1%

0%

1%

1%

2%

2%

0 0.5 1 1.5 2 2.5

Low CoBar High CoBar

post-ranking beta

Aver

age

Exce

ss R

eturn

in M

onth

s 1-6

post-ranking beta Aver

age

Exce

ss R

eturn

in Y

ear

3

Figure 6: This figure shows how the information in 𝐶𝑜𝐵𝐴𝑅 about time-variation in the expected holding and

post-holding return to beta-arbitrage strategies decays as staler estimates of beta are used to form the beta-

arbitrage strategy. At the end of each month, all stocks are sorted into deciles based on their market beta

calculated using daily returns in the past 12 months. To account for illiquidity and non-synchronous trading,

we include on the right-hand side of the regression equation five lags of the excess market return, in addition

to the contemporaneous excess market return. The pre-ranking beta is simply the sum of the six coefficients

from the OLS regression. We then compute the strategy return as the value-weight low-beta decile return

minus the value-weight high-beta decile return. We separately regress the abnormal return of the beta-

arbitrage strategy in months one-six and year three on 𝐶𝑜𝐵𝐴𝑅. In this process, we first use a fresh estimate

of beta, calculated using daily returns in the past 12 months. We then repeat the analysis using stale betas,

computed from daily returns in each of the prior 5 years (thus having different beta portfolios as of time zero

for each degree of beta staleness). We plot the corresponding regression coefficients (results for months 1-6

plotted with a blue square and results for year 3 plotted with a red circle) for each of the six beta-arbitrage

strategies, ranging from fresh beta to five years stale beta. The top (bottom) panel reports results based on

three-factor (four-factor) alpha.

Fam

a-F

rench

Alp

ha

Carh

art

Four-

Fact

or

Alp

ha

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 1 2 3 4 5

Month 1-6 Year 3

Stale beta Fresh beta

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 1 2 3 4 5

Month 1-6 Year 3

Fresh beta Stale beta